Canonical Descriptions of High Intensity Laser-Plasma ...
Transcript of Canonical Descriptions of High Intensity Laser-Plasma ...
Canonical Descriptions of High
Intensity Laser-Plasma
Interaction
B. J. Le Cornu
School of Computing, Engineering and Mathematics
University of Western Sydney
A thesis submitted for the degree of
PhilosophiæDoctor (PhD)
2014
Abstract
The problem of laser-plasma interaction has been studied extensively
in the context of inertial confinement fusion (ICF). These studies have
focussed on effects like the nonlinear force, self-focusing, Rayleigh-
Taylor instabilities, stimulated Brillouin scattering and stimulated
Raman scattering observed in ICF schemes. However, there remains
a large discrepancy between theory and experiment in the context
of nuclear fusion schemes. Several authors have attempted to gain
greater understanding of the physics involved by the application of
standard or ‘canonical’ methods used in Lagrangian and Hamiltonian
mechanics to the problem of plasma physics.
This thesis presents a new canonical description of laser-plasma in-
teraction based on the Podolsky Lagrangian. Finite self-energy of
charged particles, incroporation of high-frequency effects and an abil-
ity to quantise are the main advantages of this new model. The nature
of the Podolsky constant is also analysed in the context of plasma
physics, specifically in terms of the plasma dispersion relation. A new
gauge invariant expression of the energy-momentum tensor for any
gauge invariant Lagrangian dependent on second order derivatives is
derived for the first time. Finally, the transient and nontransient
expressions of the nonlinear ponderomotive force in laser-plasma in-
teraction are discussed and shown to be closely approximated by a
canonical derivation of the electromagnetic Lagrangian, a fact that
seems to have been missed in the literature.
Declaration
The work presented in this thesis is, to the best of my knowledge
and belief, original except as acknowledged in the text. I hereby
declare that I have not submitted this material, either in full or in
part, for a degree at this or any other institution.
(Signature)
Acknowledgements
The lion’s share of my gratitude belongs to my principal supervisor,
Dr. Reynaldo Castillo. At any given meeting, Dr. Castillo faced
either a barrage of unfocussed exuberance or grim dejectedness. I
thank him for his years of patience and instruction. My associate
supervisors, Dr. Timothy Stait-Gardner and Prof. Andrew Francis
were instrumental. Prof. Francis first inspired me to pursue science
beyond my undergraduate degree and guided me in my first real foray
into mathematics. Dr. Stait-Gardner was always available to answer
crazy and/or stupid questions, and his lessons were invaluable to me.
All of my supervisors are rare in that they distinguish themselves not
only by the breadth and depth of their knowledge, but by their ability
to impart a genuine understanding of it.
The support of my family as a whole was phenomenal. My father and
my sister Kieren have always been there, supporting me and shielding
me from reality (which often reared its head in bill form.) My mother
and Amro provided the greatest support when it was needed most,
without which I would never have finished this work. Evan and Clare
tolerated my erratic schedule and kept me solvent in the final year,
and Evan perhaps helped keep me sane. Jenna sacrificed a tremendous
amount for me in the first few years, for which I will always be in her
debt. Nidin has been an excellent sounding board and an even better
friend. Thanks also to all my friends who kept me caffeinated and in
contact with the rest of the world, especially Justin and Emma.
Contents
1 Introduction 1
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Plasma Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Debye Length . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Collision Frequency . . . . . . . . . . . . . . . . . . . . . . 13
1.2.4 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.5 Plasma Dispersion Relation . . . . . . . . . . . . . . . . . 16
1.2.6 Refractive Index . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.1 Eulerian and Lagrangian Coordinates . . . . . . . . . . . . 20
1.3.2 Particle-In-Cell Simulations . . . . . . . . . . . . . . . . . 24
1.3.3 Relevant Fluid Equations . . . . . . . . . . . . . . . . . . 24
1.4 Kinetic Plasma Models . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 Quantum Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.1 Scale of Quantum Effects in Plasmas . . . . . . . . . . . . 27
1.5.2 Need For a Quantum Plasma Description . . . . . . . . . . 29
2 Lagrangian and Hamiltonian Mechanics 31
2.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Canonical Derivation of the Maxwell Energy-Momentum Tensor . 35
2.3 Lagrange Multipliers versus Constrained Variations . . . . . . . . 38
v
CONTENTS
2.4 Canonical Derivation of the Lorentz Force . . . . . . . . . . . . . 42
2.5 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Lagrangian and Hamiltonian Formulations of Laser-Plasma In-
teraction 52
3.1 The Boltzmann-Vlasov Distribution . . . . . . . . . . . . . . . . . 53
3.2 Ideal Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Clebsch Potential Representation . . . . . . . . . . . . . . . . . . 57
3.4 Virtual Fluid Displacement . . . . . . . . . . . . . . . . . . . . . 63
3.5 Relativistic Constraints in Fluid Dynamics . . . . . . . . . . . . . 69
3.6 Maxwell-Vlasov System . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6.1 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . 76
3.7 Guiding Centre Motion . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 The Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . 80
4 The Energy-Momentum Tensor In Higher-Derivative Theories 82
4.1 The Gauge Invariant Electromagnetic Energy-Momentum Tensor 82
4.2 Lagrangians with Higher-Order Derivatives . . . . . . . . . . . . . 86
4.3 The Gauge Invariant Energy-Momentum Tensor for the Podolsky
Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 The Nonlinear Ponderomotive Force in Laser-Plasma Interac-
tion 102
5.1 The Ponderomotive Force . . . . . . . . . . . . . . . . . . . . . . 102
5.2 The Nontransient Nonlinear Ponderomotive Force . . . . . . . . . 105
5.3 The Transient Nonlinear Ponderomotive Force . . . . . . . . . . . 110
6 Concluding Remarks 117
References 121
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1
Introduction
The study of laser-plasma interaction was born of attempts to recreate something
like the Teller-Ulam Hydrogen bomb on a small, controlled scale. The Hydrogen
bomb is essentially a two-stage device, consisting of a primary fission explosion
that bombards a mixture of Deuterium and Tritium (DT) with enough energy
to create a secondary fusion reaction. The idea to use this same principle for
commercial power generation can be traced to Teller in the late 1950s, although
it was in the early 1960s that Nuckolls, Kidder, Colgate and Zabawski (all of whom
worked at Lawrence Livermore National Laboratory) first considered using lasers
to implode a small DT target, resulting in a fusion reaction [1] [2]. This method
of laser-driven fusion is referred to as inertial confinement fusion (ICF).
Nuclear fusion offers a release of energy per unit mass that cannot be beaten
by chemical or even fission reactions, and fusion energy comes without the cost
of radioactive waste or the threat of nuclear plant meltdown. However, in the
decades since the 1960s, successive predictions of the laser power required for
ignition (the point at which the fusion reaction becomes self-sustaining) have been
defied as the sheer scale and complexity of the physics has gradually been realised.
For instance, one of the many obstacles is the near-perfect spherical compression
of the fusion target that is required to reach ignition. Small irregularities in the
symmetry of the target compression become exponentially more pronounced over
the time scales involved, leading to a loss of energy and momentum delivered to
1
the centre of the imploding target.
The largest inertial confinement facility in the world is the National Ignition
Facility (NIF) located in California. In March 2012, NIF exceeded its design laser
energy output of 1.8 megajoules by delivering 1.85 megajoules in trillionths of a
second - a pulse of more than 500 terawatts at peak power. The ultimate goal of
the NIF is to demonstrate ignition, which has still never been achieved in ICF.
The NIF has failed to meet its own deadline for achieving ignition, but it has
demonstrated a net energy gain from the fusion of a DT pellet compared to the
laser energy that was delivered to it [3] (this is still not a net release compared
to the total laser energy as there are significant losses in the system before the
laser beams reach the target). All theoretical models and simulations indicated
that NIF should be able to achieve ignition comfortably and reasonably quickly
after coming on line [4], and its failure to do so suggests that a sound theoretical
understanding of the problem is still over the horizon.
An alternative to ICF is the magnetic confinement fusion (MCF) scheme.
MCF precipitates a fusion reaction by injecting plasma into a large toroidal
chamber (most commonly a tokamak) - ideally confining the plasma with electro-
magnets arranged around the chamber - while bombarding it with high energy
radiation. However, plasma turbulence is a major problem causing deconfinement
and damage to the reactor interior. Confinement time is usually on the order of
seconds and no MCF facility has achieved ignition to date. The ITER1 (‘the
way’, in Latin) project is in the early stages of construction at the time of writing
and will be the largest MCF tokamak facility in the world upon its predicted
completion in 2019.
The stark disagreement between theory and experiment in ICF and MCF may
simply be due to the enormous number of degrees of freedom that must be taken
into account - too many even for the sophisticated models that have so far been
number-crunched on supercomputers over periods of days or weeks. Alternatively,
1Originally an acronym for International Thermonuclear Experimental Reactor
2
there may be something missing from the theoretical side of physics, lost in
some order of approximation or assumption made for classical physics where
the discrepancy goes unnoticed. Of course, there is no single modification to the
theory of laser-plasma interaction that will serve as a silver bullet. However, it is
the aim of this thesis to present some modifications using ‘canonical’ methods that
may improve the accuracy of theoretical predictions made in the realm of nuclear
fusion. These methods are canonical in the sense that they arise from either a
Lagrangian or Hamiltonian description of the situation. In Lagrangian mechanics,
the equations of motion flow from an application of Hamilton’s principle via a
well-understood process. Hamiltonian formulations may arise from the usual
Legendre transformation of the Lagrangian or a more general transformation
according to Dirac’s constraint theory [5]. Perhaps the most interesting theories
are found by guessing the form of a functional based on the total energy of the
system and its degrees of freedom, entirely independent of a Lagrangian, and then
defining a ‘noncanonical Poisson bracket’ that serves to endow this functional
with a Hamiltonian structure. These canonical methods will be explored in the
context of ICF within this thesis. While nuclear fusion is the main focus of all
research into laser-plasma interaction, the subject is more general than that and
really applies to the physics of any charged particles in the presence of ultra-high
intensity electromagnetic radiation.
There are five principal results presented in this thesis. First, a derivation of
a gauge invariant energy-momentum tensor for a Lagrangian with second-order
derivatives of the electromagnetic potentials is presented [6]. Second, a proof
that the aforementioned tensor reproduces a tensor found by Podolsky [7] for his
theory of generalised Maxwellian electrodynamics is given [8]. Third, Podolsky
electrodynamics is applied to the case of laser-plasma interaction, resulting in
the derivation of a new plasma dispersion relation [8]. Fourth, an expression
closely approximating the nonlinear ponderomotive force, first applied by Hora
[9] to laser-plasma interaction, is derived using Lagrangian mechanics [10]. Fi-
3
nally, a refutation of a result due to Rowlands [11] that attempted to generalise
the transient nonlinear ponderomotive force [12] is put forth, along with a new
alternative generalisation [10].
The remainder of this chapter is devoted to a summary of some basic principles
and equations applied to plasma physics. This summary is not exhaustive, but
is designed to introduce the equations and concepts that will be required later in
a way that ensures this thesis is self contained.
Chapter 2 is a review of all the concepts in Lagrangian and Hamiltonian
mechanics that are necessary for the results presented later. Nothing new is
presented in this chapter, although several derivations are presented that are
not found in standard textbook treatments. These derivations have a twofold
purpose; to familiarise the reader with variational principles and to demonstrate
the general method that will be applied later to achieve new results, so that the
reader can be convinced of their correctness.
Chapter 3 comprises a review of the literature related to the topic of varia-
tional methods in plasma physics and, more generally, in fluid mechanics. The
variational method considered by all these authors is Hamilton’s principle. Some
of the literature specifically addresses the case of laser-plasma interaction, in ei-
ther a Lagrangian or Hamiltonian description, or sometimes both. Within this
chapter, §3.5 contains a critical review of one attempt to apply Hamilton’s prin-
ciple to the case of a relativistic perfect fluid, which does include a minor original
contribution on the part of this author.
Chapter 4 contains the first of the original results presented in this thesis,
namely the derivation of a new gauge invariant energy-momentum tensor for any
gauge invariant Lagrangian dependent on second order derivatives of the elec-
tromagnetic potentials. The Podolsky Lagrangian is then substituted into this
new tensor, yielding an expression that is manifestly gauge invariant without
further tinkering. The equivalence of this expression to Podolsky’s original ten-
sor is then demonstrated. Finally, Podolsky electrodynamics is presented as a
4
1.1 Notation
possible candidate for tackling problems in laser-plasma interaction, due to its
elimination of the infinite electron self-energy and ability to account for higher
frequency phenomena. A new plasma dispersion relation is derived using the
Podolsky equations of motion.
Chapter 5 presents a discussion of the nonlinear ponderomotive force as ap-
plied to laser-plasma interaction. This thesis then points out that derivations that
led to the force expression could probably be replaced by one simple application
of Hamilton’s principle to a slightly modified version of the standard Maxwell
electromagnetic Lagrangian. The transient expression of the nonlinear pondero-
motive force is then briefly discussed, before this author presents an alternate
method of generalising this expression to one that is Lorentz invariant and uses
four-dimensional notation, in contrast to an expression derived by Rowlands [11]
that this author believes to be incorrect.
Chapter 6 contains some concluding remarks that compare the original results
in this thesis with the rest of the literature that was reviewed and presents a
hopeful picture of their application to future experiments in nuclear fusion.
1.1 Notation
All units are SI throughout this thesis. Greek indices will always range from 0
to 3 and Latin indices from 1 to 3 unless otherwise indicated. The summation
convention is employed throughout; a repeated index in any term of an expression
is summed over all possible values, for example,
aiai = a1a
1 + a2a2 + a3a
3.
These indices are known as dummy indices and must always appear in co-
variant/contravariant pairs (respectively lower/upper indices, although there is
no distinction between contravariant and covariant components in 3-dimensional
Cartesian coordinate systems). Free indices appear exactly once in every term
5
1.1 Notation
of an expression, identifying a particular component of a vector or tensor. For
instance, the index i is the free index in the expression ai = gijaj and denotes
the ith component of the vector a. The relationship between covariant and con-
travariant components is determined by the relevant metric tensor gij, so that
ai = gijaj and ai = gijaj. The Minkowski metric used throughout this thesis for
relativistic problems is defined as
gµν =
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
.
The infinitesimal squared spacetime interval is given by
ds2 = dxαdxα = c2dτ 2
where τ is proper time and c the speed of light. The covariant spacetime coordi-
nates are xµ = (ct,−x,−y,−z) which implies that the four-velocity components
are
uµ = dxµ/dτ = (cγ,−vxγ,−vyγ,−vzγ)
where the Lorentz factor is γ = 1/√
1− v2/c2. The covariant components of the
four-gradient are
∂µ =
(∂
∂x0,− ∂
∂x1,− ∂
∂x2,− ∂
∂x3
)=
(∂
c∂t,−∇
).
Comma subscripts always indicate the partial derivative with respect to co-
ordinates xµ, that is,
∂f
∂xµ:= f,µ and
∂f
∂xµ:= f ,µ.
For more complicated tensor expressions, any index appearing to the right of
6
1.1 Notation
the comma denotes a derivative regardless of whether the index is upper or lower
relative to the comma, i.e.
Fαβ,γγ =
∂2Fαβ
∂xγxγ.
The electromagnetic four-potential has covariant components
Aµ = (φ/c,−Ax,−Ay,−Az)
where φ is the scalar potential and A the vector potential. The relationship
between the electric and magnetic fields is given in terms of these potentials such
that
E = −∇φ− ∂A
∂tand B = ∇×A
or, equivalently in component form,
Ei = −φ,i −∂Ai∂t
and Bi = εijkAk,j
where the Levi-Civita tensor is given by
εijk =
1 if ijk = 123, 312 or 231
−1 if ijk = 132, 213 or 321
0 if i = j, j = k or i = k.
The covariant form of the electromagnetic tensor Fµν := Aν,µ − Aµ,ν can be
expressed explicitly in matrix form as
Fµν =
0 Ex/c Ey/c Ez/c
−Ex/c 0 −Bz By
−Ey/c Bz 0 −Bx
−Ez/c −By Bx 0
7
1.2 Plasma Parameters
or, for the contravariant,
F µν = gαµgβνFαβ =
0 −Ex/c −Ey/c −Ez/c
Ex/c 0 −Bz By
Ey/c Bz 0 −Bx
Ez/c −By Bx 0
.
The electromagnetic tensor satisfies the Jacobi identity such that
Fαβ,γ + Fβγ,α + Fγα,β = 0.
The vacuum permittivity and permeability constants are ε0 and µ0 respec-
tively, defined by the relation
c2 =1
µ0ε0.
The Poynting vector, which represents the directed electromagnetic energy
flux density is
S =1
µ0
E×B.
Planck’s constant h in SI units is
h ≈ 6.626× 10−34Js
and the reduced Planck’s constant is
~ =h
2π≈ 1.055× 10−34Js.
1.2 Plasma Parameters
While plasma is often referred to as the ‘fourth state of matter’, there is no exact
definition of a plasma. A plasma is best described qualitatively as “an ionized
gas whose behaviour is dominated by collective effects and by possessing a very
8
1.2 Plasma Parameters
high electrical conductivity” [13]. There are no exact quantitative criteria that
a gas must meet in terms of temperature, pressure or level of ionisation for it to
be considered a plasma. An ionization level of less than 1% can be sufficient for
a gas to display plasma behaviour. The key distinguishing feature of a plasma
compared to a gas is the collective behaviour of its constituent particles, governed
as they are by the electromagnetic force. In a gas of electrically neutral particles,
interactions are governed by thermokinetics and collisional forces which operate
over shorter distances, essentially at the speed of sound in the gas. The unique
behaviour of a plasma is due to the propagation of electromagnetic influences
throughout an ionised gas at the speed of light.
Plasma physics is a broad discipline that necessarily encompasses fluid dynam-
ics, thermodynamics, relativity and even quantum mechanics, depending on the
mathematical model being employed and the temperatures and densities being
considered. For instance, it has been speculated that Jupiter’s core may consist
of liquid metallic Hydrogen - a theoretical state of matter that can only occur at a
phenomenally high pressure - a state of matter that probably requires a quantum
plasma description. The acceleration of charged particles to near light speeds
in the region of pulsars or active galactic nuclei requires a relativistic plasma
description [14].
If a plasma is modelled as a continuous fluid, then the laws of thermodynamics
are necessary for its complete description. Plasmas are further characterised
according to whether they are ‘hot’, ‘cold’ (high or low degree of ionisation), or
collisional.
Despite the ambiguity in defining a plasma, there are several quantitative pa-
rameters and characteristics that are used to describe only plasmas, aside from the
usual parameters of temperature, pressure, density etc. These will be discussed
in this section.
9
1.2 Plasma Parameters
1.2.1 Plasma Frequency
For the purposes of this thesis, the plasma frequency refers specifically to a fre-
quency related to the ionized electrons in a plasma, since their movement within
the plasma is much faster and over much larger distances compared to the sig-
nificantly heavier ions. Exotic plasmas such as electron-positron plasmas or the-
oretical quark-gluon plasmas will not be considered here. Plasmas have a char-
acteristic frequency because the electrons are not really free from the ions even
once stripped from their orbits. The plasma frequency can be derived in several
slightly different ways using: assumptions about the AC shielding of plasmas
[13]; solving the set of equations given by Gauss’ law with the continuity and
Euler equations by linearising them [15, 16, 17], or perhaps more simply from
arguments about perturbations in electron density [18]. The following derivation
follows the same basic argument put forth by Hora [18].
Consider electrons in equilibrium against a static background of positively
charged ions whose mass can be assumed infinite for all practical purposes. If the
electron density is perturbed by a small amount in space, then this variation in
the density is
δn = − ∂n∂xi
δxi (1.1)
and the disturbance creates an electric field where there was none before. Gauss’
law then gives
Ei,i =
q
ε0δn
where q and n are the electron charge and electron number density respectively.
Using Eqn (3.34), this can be expressed as
∂
∂xi
(Ei +
q
ε0nδxi
)= 0.
This implies that the expression in the brackets above is a constant, although
since the perturbation in the density function is creating the electric field (the
10
1.2 Plasma Parameters
plasma was assumed neutral to begin with), when δxi = 0 it is also the case that
E = 0 and therefore
Ei = − qε0nδxi. (1.2)
The Lorentz force on an electron whose position was perturbed by the in-
finetismal amount δx when the density was perturbed, is now given by
d2δxi
dt2=
q
mEi.
Using Eqn (1.2), this can be rewritten as
d2δxi
dt2= − q
2n
ε0mδxi
which suggests a solution of the form eiωpt where ωp is identified as the plasma
frequency such that
ωp =
√q2n
ε0m, (1.3)
or, given the best current values for the electron mass, electron charge and electric
permittivity,
ωp = 56.415(m3/2s−1)√n.
The above expression is of course not valid if relativistic effects are enough to
appreciably alter the mass of the electron. The preceding derivation of the plasma
frequency didn’t require any assumptions about or knowledge of a plasma. All
that was referred to was a perturbed density function (although this could be
a density function for any collection of charged particles) and Gauss’ law. This
makes sense, since any electron pushed away from an equilibrium it managed to
reach in the presence of a background electric field will move back and forth in
simple harmonic motion until dissipative forces eventually return it to equilib-
rium.
The plasma frequency is key to understanding how electromagnetic waves
travel through a plasma. When the refractive index of a plasma is derived in
11
1.2 Plasma Parameters
§1.2.6, it will become obvious that any light or electromagnetic wave with a fre-
quency less than the plasma frequency will be absorbed or reflected rather than
transmitted by the plasma. Radio waves in certain frequencies can be transmit-
ted between two points on the surface of the Earth without line of sight as the
ionosphere is (as the name suggests) partially ionised and has a characteristic
plasma frequency. The level of ionisation in the upper atmosphere depends on
the time of day, level of solar activity and other considerations.
1.2.2 Debye Length
The Debye length is a fundamental unit of distance over which all significant
plasma interactions are measured. Consider an ion injected into a plasma in
which the electrons and ions are initially equidistant and static. The ion then
creates a disturbance whereby the negatively charged electrons will be attracted
toward the ion and the other ions will be repelled. This results in a cloud of
negative electrons surrounding the positive ion that partially negates the electric
field of the ion, that is, the ion is ‘shielded’ from the rest of the plasma. The
distance over which this shielding occurs is in a plasma is the Debye length. The
derivation of an expression for this length can be done using the equations of
fluid dynamics [13] the balancing of thermal and electric forces [19]. However,
this author puts forward Hora’s [18] much simpler argument for deriving the
Debye length.
Since the plasma frequency provides an inverse time scale over which mean-
ingful phenomena occur within the plasma, the distance over which electrons can
propagate an electrical influence in the plasma is limited by their velocity divided
by the plasma frequency.
According to the law of equipartition [20], the kinetic energy of a particle is
related to the temperature of it species (assuming that the plasma is close enough
to thermal equilibrium to allow a well-defined temperature for its species) such
12
1.2 Plasma Parameters
that
1
2m(v2x + v2y + v2z) =
3
2kBT
where kB is Boltzmann’s constant and T is the electron temperature. In terms
of the magnitude of the velocity averaged over three dimensions, this gives
vavg =
√kBT
m.
Therefore the Debye length λD is
λD =vavgωp
=
√ε0kBT
q2n. (1.4)
Collective behaviour unique to plasmas only occurs over distances larger than
the Debye length. Therefore, an important factor to consider in plasma physics
is the number of particles within a certain volume with dimension on the order of
the Debye length. Consider a sphere of radius λD, then the number of particles
Λ within this sphere is given by its volume multiplied by the density function n
[13]:
Λ =4
3πλ3Dn. (1.5)
1.2.3 Collision Frequency
The collision frequency of a plasma refers to the number of collisions per unit
time of particles within the plasma. However, in many circumstances, the electron
temperature (and therefore average speed) is much higher than that of the plasma
ions, and the electron collision frequency is the dominant collision frequency.
A collision is considered to be a strong one when the incident particle’s kinetic
energy is comparable to the Coulomb potential between itself and the particle it
collided with [13]. That is,
1
2mv2 ≈ |qQ|
4πε0rcoul
13
1.2 Plasma Parameters
where rcoul is the distance between the particles and Q is the charge on the
particle being collided with, which may be an ion with charge −q or another
electron with charge q. Since only the absolute value is considered, this will be
simplified as |Qq| = q2. If long-range effects are to dominate over short-range
Coulomb interactions, then the average distance between particles in the plasma
ravg must be much greater than the distance at which the Coulomb potential is
comparable to the kinetic energy rcoul. It is common to approximate the velocity
according to the relation
mv2 ≈ kBT,
in contrast to
ravg >>q2
2πε0mv2=
q2
2πε0kBT.
The collision cross section for an electron-electron collision is then given by
σee = πr2coul =q4
4πε20k2BT
2
and the collision frequency must be given by this cross section multiplied by the
particle density and velocity,
νee = σeenv =nq2
4πε20k3/2B m1/2T 3/2
=1
3
ωpΛ.
1.2.4 Ohm’s Law
A version of Ohm’s law can be derived for a plasma using the two-fluid equations.
The following derivation, especially for a collisional plasma, follows Hora [18]. The
two-fluid equations are
minidvidt
= −∇(nikBTi)− ZniqE− niZq
cvi ×B−miniνei(vi − ve) (1.6)
14
1.2 Plasma Parameters
for the ion fluid and
menedvedt
= −∇(nekBTe) + neqE + neq
cve ×B−meneνei(vi − ve) (1.7)
for the electron fluid. Recall that since the velocities are for a fluid, the total
time derivative represents the material derivative (see §1.3.1) such that
d
dt=
∂
∂t+ v · ∇v.
The ion and electron fluids will be considered to initially be in a certain state
for which the two fluid equations are exactly solvable. Disturbances created by
waves travelling through the plasma will be ‘perturbed’ solutions to Eqns (1.6)
and (1.7), approximated to first order so that n = n(0) + n(1) and v = v(0) + v(1)
with the first order terms proportional to ei(k·r−ωt). Several assumptions will be
made, namely that the plasma is: collisionless (νei = 0); cold (∇nkBT = 0);
homogeneous (n(1) = 0); without magnetic field (B = 0); and without drift
velocity (v(0) = 0). Eqns (1.6) and (1.7) now become
mini(0)∂vi(1)∂t
= −Zni(0)qE(1) (1.8)
=⇒ imini(0)ωvi(1) = Zni(0)qE(1)
for the ions and
mene(0)∂ve(1)∂t
= ne(0)qE(1) (1.9)
=⇒ imene(0)ωve(1) = −ne(0)qE(1)
for the electrons. Note that Eqns (1.8) and (1.9) combined give
vi(1) = −Zme
mi
ve(1)
and since mi >> me, it follows that the ion velocity is negligible and can be
15
1.2 Plasma Parameters
ignored for practical purposes. This leaves the first order equation (1.9) for the
electron fluid velocity, which can be expressed as
ne(0)ve(1) = ine(0)q
meωE(1)
or, recalling Eqn (1.3),
Je = iε0ω2p
ωE(1). (1.10)
Eqn (1.10) is a new generalised Ohm’s law for a collisionless plasma. A col-
lisional version can also be obtained under different assumptions, which will be
discussed later in §5.2. For now, it is enough to give the version of Eqn (1.10)
generalised to include collisions:
dJ
dt+ νJ = ε0ω
2pE(1).
Solving this where J ∝ e−i(k·r−ωt) gives
J =ε0ω
2p
iω(1− iν/ω)E, (1.11)
the collisional version of Ohm’s Law for a plasma.
1.2.5 Plasma Dispersion Relation
Since the dispersion relation for a plasma governs the way in which waves prop-
agate through a plasma, a fluid model of the plasma is necessary to derive it.
Several authors give derivations of the plasma dispersion relation [15, 16, 17],
and while they all use similar arguments, this derivation more closely follows
Somov’s [15].
Consider Eqn (1.10), the generalised Ohm’s Law derived in §1.2.4. Recall
Faraday’s law and Ampere’s Law,
∇× E = −∂B
∂t, (1.12)
16
1.2 Plasma Parameters
∇×B = µ0J +1
c2∂E
∂t, (1.13)
and assume the first order electric field contribution E(1) is also proportional to
ei(k·r−ωt), then substitute Eqn (1.10) into (1.13);
∇×B(1) = iµ0ε0ω2p
ωE(1) −
iω
c2E(1).
Taking the time derivative of both sides and then substituting Eqn (1.12)
gives
−∇× (∇× E(1)) =ω2p
c2E(1) −
ω2
c2E(1). (1.14)
Consider any particular component of the left hand side of the above equation,
say, the x-component, then
∇× E = (ikyEz − ikzEy, ikzEx − ikxEz, ikxEy − ikyEx)
=⇒ (∇× (∇× E))x =∂(ikxEy − ikyEx)
∂y− ∂(ikzEx − ikxEz)
∂z
= −kxkyEy + k2yEx + k2zEx − kxkzEz
However, since the wave vector k points in the direction of propagation of the
electromagnetic wave, and the electric field is perpendicular to this direction, it
is the true that
k · E = 0 =⇒ kxEx = −kyEy − kzEz
and therefore
(∇× (∇× E))x = k2Ex.
Substituting this into Eqn (1.14) gives
−k2Ex(1) =
(ω2p
c2− ω2
c2
)Ex(1)
17
1.2 Plasma Parameters
and the plasma dispersion relation is therefore
ω2p + c2k2 = ω2. (1.15)
Another way of expressing the dispersion relation serves to highlight the cutoff
frequency below which waves cannot propagate through the plasma;
ck
ω=
√1−
ω2p
ω2.
It is clear from the above expression that any wave with frequency ω ≤ ωp
will have a wave vector that is zero or imaginary. An imaginary wave vector
corresponds to reflection of the wave by the plasma. Since the only variable in
the plasma frequency expression is the density, this problem could be rephrased
to find a critical density of the plasma, above which a wave of frequency ω cannot
penetrate. According to Eqn (1.3),
ne =ε0meω
2p
q2.
Since ω ≥ ωp to avoid being cutoff, the critical density nc can be defined as
nc =ε0meω
2
q2.
If the plasma electron density is equal to or greater than nc, the wave will be
cutoff.
1.2.6 Refractive Index
The refractive index η for any medium is defined by
η =c
vphase
where c is the speed of light and vphase the phase velocity of light in the
18
1.2 Plasma Parameters
medium. It is straightforward to derive the refractive index for a collisionless
plasma given the dispersion relation in Eqn (1.15). First, the phase velocity can
be expressed in terms of the frequency and magnitude of the wave vector such
that
vphase =ω
k.
For a plasma, the refractive index is then, from Eqn (1.15),
η =ck
ω=
√1−
ω2p
ω2.
This can be easily adapted for a collisional plasma using the collisional Ohm’s
law in Eqn (1.11). Exactly the same procedure that led to Eqn (1.15) (subsi-
tuting E for J in Ampere’s Law using Ohm’s law and simplifying via Fourier
transformation) can be used to give a collisional dispersion relation using Eqn
(1.11) such thatω2p
(1− iν/ω)+ c2k2 = ω2 (1.16)
and the refractive index is then
η =ck
ω=
√1−
ω2p
ω2(1− iν/ω). (1.17)
It will also be useful later to note that the refractive index is related to the
permittivity and permeability of the medium in question. In free space
c =1
µ0ε0,
and for an electromagnetic wave propagating in a medium with phase velocity
vphase, this becomes
vphase =1√µε
where µ and ε are the permittivity and permeability constants. The auxiliary
fields D and H in matter are then defined in terms of the vacuum electric and
19
1.3 Fluid Mechanics
magnetic fields, E and B:
D = ε0εrE = εE
and
H =1
µ0µrB =
1
µB,
where εr and µr are the dimensionless relative permittivity and permeability
constants, respectively. Therefore,
η =c
vphase=
√µε
µ0ε0=√µrεr. (1.18)
1.3 Fluid Mechanics
Plasma phenomena are typically investigated using one of two broad classes of
models - kinetic models and fluid models. Kinetic models will be discussed briefly
in §1.4. A fluid model describes plasma by looking at its bulk properties as a
fluid, or as two fluids (an electron fluid and an ion fluid) which then allows the use
of all the principles and equations of thermodynamics. A fluid description makes
use of the roughly Maxwell-Boltzmann distribution of particles in collisionless
plasmas. This is obviously a macroscopic idealisation of the physics and such an
approximate description cannot account for certain microscopic phenomena like
plasma double layers.
The material presented in this section is a brief summary of the elementary
concepts and equations of fluid mechanics, which can be found in any textbook
on the subject, e.g., [21] [22].
1.3.1 Eulerian and Lagrangian Coordinates
Fluid mechanics can be described in two equivalent types of coordinate systems -
Eulerian and Lagrangian. In an Eulerian description, the volume under consider-
ation is subdivided into static cells through which the fluid flows. The dynamics
of the system are then described by vectors attached to each cell which change in
20
1.3 Fluid Mechanics
direction and magnitude over time. In a Lagrangian description, fluid particles
are tracked individually as they move throughout a volume over time (one might
practically think of dropping coloured dye or a neutrally buoyant object of negli-
gible mass into the fluid which then stays with a fluid particle as it moves). The
use of the term ‘fluid particle’ and the mathematical description of a fluid as a
continuous system is an approximation. Below the microscopic scale, a fluid is of
course made of molecules and so cannot be defined at every point in space.
An Eulerian position vector will be denoted by x and Eulerian velocity is
v(x, t). The vector position of a fluid particle in the Lagrangian description will
be given by ξ(a, t) where a is the initial position of the fluid particle and t is
time. The initial position a is the label that uniquely identifies each fluid particle
as it moves throughout space. Consistent with this notation, the Eulerian fluid
velocity vE can be given in terms of the Lagrangian particle velocity vL such that
vE(x, t)|x=ξ(a,t) =∂ξ(a, t)
∂t= vL(a, t)
which simply states that the Eulerian velocity at any point in space at a certain
time is the same as the instantaneous velocity of a Lagrangian particle passing
through that point at that time. More generally, any physical quantity carried
along by the fluid must have its time derivative given by
d
dt=
∂
∂t+ v · ∇, (1.19)
which is the material derivative. The material derivative gives the time derivative
holding position constant (the first term) plus the derivative holding time constant
while moving through the vector field (the second term). It is simply the total
derivative of something that depends on both space and time. The acceleration
of the fluid is given by the material derivative of the fluid velocity
dv
dt=∂v
∂t+ v · ∇v.
21
1.3 Fluid Mechanics
If the fluid velocity does not depend on time then it is a steady flow. If it does
not depend on space then the velocity has the same magnitude and points in the
same direction everywhere, which is an isotropic flow (the usage of this term may
differ from other authors). As an example, consider a (fairly unusual) vector field
defined by v = (x + y + t/2, x − y + 2t) where t is a dimensionless parameter
representing time. This vector field is neither steady nor isotropic. This can be
visualised by a stream plot, where a vector is attached to each point in space
indicating the magnitude and direction of the fluid velocity at that point.
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Figure 1.1: The vector field v = (x + y + t/2, x− y + 2t) at t = 0, 2
The lines formed by the vectors joined tip to tail are referred to as the fluid
streamlines. If the flow were steady, then the streamlines in the plot above on
the left would coincide with the pathlines - the trajectories of Lagrangian fluid
particles. However, with a dependence on time, Fig. 1.1 shows the vector field
drifting away from the origin in the ‘northwest’ direction so the motion of any one
fluid particle is more complicated over time. If a coloured dye were continually
being injected at a certain point, then as the vector field changes in time, the dye
would leave a streakline trailing the leading point of the pathline of the first drop
of dye injected.
To calculate the pathlines of a single particle placed in the fluid, consdier that
22
1.3 Fluid Mechanics
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Figure 1.2: The pathline of a single fluid particle in the steady vector field v =(x + y, x− y)
the velocity vector at any point is parallel to direction in which a particle is in-
finitesimally pushed at that point. So, the expressions for velocity and differential
position vector are
v = (vx, vy),
dr = (dx, dy).
Since these vectors are parallel, it is true that
vxdx
=vydy.
For v = (x + y + t/2, x − y + 2t), the above expression can be integrated to
give
x2
2− y2
2− 2xy + t
(2x− y
2
)= C (1.20)
where C is a constant with values corresponding to the pathlines of different fluid
particles. It can be seen from Eqn (1.20) that in the static case (t = 0), the
pathlines follow a hyperbolic orbit, which is to be expected from looking at the
streamlines in Fig. 1.1 and the pathline in Fig. 1.2. In the unsteady case, as t
23
1.3 Fluid Mechanics
increases, the linear expression 2x− y/2 begins to dominate over the hyperbolic
part of the equation and so the pathlines become flatter and flatter. This can also
be seen by considering that the vector field is drifting away from the origin at a
linear rate, and the further away a particle is from the ‘centre’ of this particular
field, the less curved the streamlines become.
1.3.2 Particle-In-Cell Simulations
‘Particle-in-cell’ (PIC) refers to a method of analysing fluid (and plasma) dynam-
ics. PIC involves solving equations for both the fluid particles in continuous space
using Lagrangian coordinates and the density and current in terms of discrete Eu-
lerian ‘mesh’ points. That is, the continuously variable Lagrangian coordinates
of the fluid particles are overlaid by a grid of cells of finite size, with each cell
corresponding to certain fluid velocity, density etc. This method lends itself to nu-
merical simulations requiring a fairly large degree of computer processing power.
Simulations using a higher density of cells are of course more accurate, but all
simulations are subject to a degree of inaccuracy from the discrete nature of the
Eulerian part of the system.
Numerical simulations using ‘particle-in-cell’ techniques have been explored
since the 1950s [23]. While this thesis focuses on analytical methods of studying
laser-plasma interaction and will not discuss PIC further, it is worth noting that
PIC simulations are still very much an active and fruitful area of research [see
24, 25, 26, 27, 28].
1.3.3 Relevant Fluid Equations
Where there are no gravitational, electromagnetic, or other forces at play, New-
ton’s second law gives the force (mass density n times acceleration) as the negative
gradient of the pressure P ;
ndv
dt= −∇P. (1.21)
24
1.4 Kinetic Plasma Models
Eqn (1.21) is known as Euler’s equation. In general, the fluid density will
depend on space and time, as will the pressure. Recalling Eqn (1.19), Euler’s
equation can also be written as
∂v
∂t= − 1
n∇P − v · ∇v.
Another important equation in fluid dynamics is the continuity equation,
∂n
∂t+ v · ∇(nv) = 0. (1.22)
The continuity equation states that mass is conserved provided there are no
sources or sinks of the fluid in the volume under consideration. The product nv
is referred to as the mass flux density. If there are sources or sinks, then the right
hand side of Eqn (1.22) is nonzero.
If the fluid entropy S is constant in time, then the fluid is said to be isentropic
and
dS
dt=∂S
∂t+ v · ∇S = 0. (1.23)
Analogous to the continuity equation (1.22), the conservation of entropy flux
density can be expressed as
∂nS
∂t+ v · ∇(nvS) = 0.
While fluid mechanics is a vast topic inclusive of complicated subjects like
turbulence, the principles discussed in this section will be sufficient to complement
the discussion of variational principle in plasma physics in this thesis.
1.4 Kinetic Plasma Models
Kinetic plasma models tend to rely on numerical techniques involving a large
amount of computational processing, as opposed to fluid models which simplify
25
1.4 Kinetic Plasma Models
the situation enough for an analytical study of plasma physics. A kinetic model
may be one in which single particles are individually considered in the plasma,
each governed by the Lorentz force equation. In this case, there is no error due to
assumptions that the plasma is a continuous object in space. The single particle
model is easily the most computationally intensive method of analysing plasma
behaviour, as thousands upon thousands of equations must be solved for each
time step considered in the model if a plasma of realistic size is to be considered.
A kinetic model relying on some distribution function is less accurate than
the single particle model, but still more accurate than a fluid model, provided
the system of equations is solvable. Given a distribution function f(x,v, t), the
probability of finding a particle at time t within an infinitesimal volume x + dx
having velocity in the range v + dv is given by [20]
f(x,v, t)dxdv.
A complete description of the plasma can be found by solving the Boltzmann
or Vlasov equations coupled to the equations of electromagnetism. The Vlasov
equation is [29]
∂f
∂t+ v · ∂f
∂x+ v · ∂f
∂v= 0. (1.24)
Analogous to Eqn (1.19), the Vlasov equation is really just the total derivative
of a function that depends on space, time and velocity, which is equal to zero
when the distribution is conserved in time. If the particles being described by
the distribution function f are charged, then the acceleration can be substituted
by an expression obtained from the Lorentz force such that
∂f
∂t+ v · ∇f +
q
m(E + v ×B) · ∂f
∂v= 0.
The Vlasov equation is really just a collisionless version of the Boltzmann
26
1.5 Quantum Plasmas
equation,
∂f
∂t+ v · ∂f
∂x+ a · ∂f
∂v=
(∂f
∂t
)collisioinal
, (1.25)
where the term on the right hand side of the equation gives the contribution to
the dynamics from collisions of particles (the discovery of the Boltzmann equation
actually preceded the Vlasov equation by more than half a century).
This use of kinetic models will be discussed further in Chapter 3, in the context
of canonical formulations of plasma physics.
1.5 Quantum Plasmas
This section briefly explores the conditions under which quantum effects become
relevant in the description of a plasma. The literature on this subject is fairly ex-
tensive, and the question has been considered by many since the salient principles
of quantum mechanics began to be understood. Among the most notable early
contributions to the understanding of quantum plasmas came from a series of pa-
pers published by Bohm and Pines [30, 31, 32, 33]. Pines also authoured a review
on the subject [34] and several modern reviews are available [see 35, 36, 37].
The question addressed here is: at which densities and temperatures will a
plasma display behaviour that can only be described in the framework of quantum
mechanics? The most important quantum parameters to consider in this case are
the Fermi energy, de Broglie wavelength and spin of plasma particles. In the
case of spin, the magnetization and collision frequency of the plasma determine
whether or not spin contributes in any appreciable way to the plasma dynamics.
1.5.1 Scale of Quantum Effects in Plasmas
Quantum effects are generally thought to be relevant in plasmas of relatively high
density and low temperature. This can be made precise either in terms of the
de Broglie wavelength λD and density n of particles, or in terms of the Fermi
temperature TF = ~2(3π)2/3n2/3/2mkB and thermal temperature T (TFkB is the
27
1.5 Quantum Plasmas
Fermi energy EF ) of the plasma. In terms of these parameters,
nλ3D ≥ 1 OR TF ≥ T
give the transition point at which quantum degeneracy becomes relevant [35].
This is intuitive since if the de Broglie wavelength of particles are larger than
the distance between particles, then it would be expected that they must share
certain quantum numbers (although at least one such number must differ between
them due to the Pauli exclusion principle).
However, the notion that quantum effects are relevant only in relatively high
density/low temperature plasmas was challenged by Brodin et al. [38] who found
that in a two-fluid plasma model in which the electron species is further subdi-
vided into two classes defined by spin, quantum effects are important even in mod-
erate density/high temperature plasmas. Spin flips can be induced in electrons
by collisions or a changing external magnetic field, provided that the magnetic
field varies faster than the inverse electron cyclotron frequency ωce. Therefore,
the time scale t considered in the two-fluid plasma model with electron spin is
1
ωce< t <
1
νe
where νe is the electron collision frequency. In such a case, Maxwell’s equations
in the plasma become
∇ · E =qini − q(n+ − n−)
ε0
∇ ·B = 0
∇×B = µ0 (j +∇× (M+ + M−)) +1
c2∂E
∂t
∇× E = −∂B
∂t
for species of particles a = i,+,− (respectively ions, spin up electrons, spin down
electrons), charge qa, spin sa and magnetization M± = −2µBn±s±/~. The Bohr
28
1.5 Quantum Plasmas
magneton is µB = q~/2me where q and me are the electron charge and mass. The
equations of motion for the plasma particles are
∂na∂t
= −∇ · (nava),
mana(∂
∂t+ va · ∇)va = qana(E + va ×B)− dp
dna∇na
+2µana~
sja∇Bj +~2na2ma
∇
(∇2n
1/2a
n1/2a
),(
∂
∂t+ va · ∇
)sa = −2µa
~B× sa.
Recall that the Einstein summation convention is employed only when dummy
indices appear in upper/lower pairs in each term of an equation; however the
indices s denote only the species of particle and the total expressions require
a sum over all species. The second equation above is the Lorentz force law
with additional terms arising due to the Fermi pressure and spin. Brodin et al.
considered the case of a low frequency Alfven (ion) wave parallel to an external
magnetic field and showed that spin effects are relevant when µB√µ0ρ0/mi ≥ 1,
where ρ0 is the unperturbed density function existing prior to the application
of the external magnetic field. However, spin effects can still be suppressed if
µ0B0/kBTe << 1, in which case the thermal pressure dominates the dynamics.
1.5.2 Need For a Quantum Plasma Description
Experiment at the National Ignition Facility (NIF) in California are currently
charting new territory in condensed matter physics. NIF is capable of studying
matter compressed to the order of thousands of times that of lead under normal
conditions and at pressures on the order of Tera Pascals. The properties of matter
at such high densities are not well understood but is important to astrophysicists
as much as nuclear fusion scientists. Metallic Hydrogen is created at pressures
high enough to force the atoms to occupy space within each other’s Bohr radii,
and is thought to exist in the cores of Jupiter and Saturn. In 1996 physicists from
29
1.5 Quantum Plasmas
Lawrence Livermore National Laboratory (NIF is also located inside the LLNL)
reported the creation of metallic Hydrogen [39] for roughly a microsecond. Most
recently, Eremets and Troyan reported the creation of liquid metallic Hydrogen
and Deuterium below 300 GPa [40]. The quantum properties of the most basic
element at high densities must be understood by physicists in general, and may
be important in developing inertial confinement fusion.
A 2011 report emerging from a National Nuclear Science Administration and
Office of Science workshop highlights, among other things, the theoretical predic-
tions regarding dense matter that are yet to be confirmed but which theoretically
lie within reach of the NIF [41]. These predictions include: the creation of solid
metallic Hydrogen; a plasma phase transition in the fluid phase for Hydrogen and
Helium; a maximum of the melting curve; the melting of Hydrogen at T = 0 K;
a Wigner crystal state for Hydrogen; a superconductor and/or superfluid phase
of Hydrogen. A better theoretical understanding of the nature of quantum plas-
mas will be crucial in any further study of the unique properties of ultra-high
condensed matter.
30
2
Lagrangian and Hamiltonian
Mechanics
2.1 Lagrangian Mechanics
This chapter introduces the basic principles of Lagrangian and Hamiltonian me-
chanics. A review of the application of these ideas to the case of fluid mechan-
ics and laser-plasma interaction will be presented in Chapter 3. Discussions of
Hamilton’s principle and the Euler-Lagrange equations can be found in any good
classical mechanics textbook [e.g. 42, 43].
Lagrangian mechanics is a desirable framework in which to study plasma dy-
namics (and indeed many other areas of physics) due to the natural appearance of
the equations of motion and conservation laws from well understood operations on
a single functional - the Lagrangian. The application of Lagrangian mechanics to
plasma physics originated with Low’s Lagrangian formulation of the Boltzmann-
Vlasov equation [44]. The most notable extension of this theme in describing the
complexities of plasma physics was the noncanonical Hamiltonian formalism used
by Littlejohn in the context of guiding centre motion in magnetic confinement
fusion [45, 46]. Much work has been done in this field since these early efforts
and Hamiltonian and Lagrangian mechanics are of continuing interest to those
31
2.1 Lagrangian Mechanics
studying plasma physics and fluid dynamics in general [see 28, 47, 48, 49].
Lagrangian mechanics is an elegant formulation of classical mechanics that
allows the equations of motion to be derived from the principle that the path an
object takes will always be the path that gives a stationary value of the action.
This is known as Hamilton’s Principle, or sometimes the Principle of Least Action
(although this is a misnomer, as the action need only be an extremal value, not
necessarily a minimum). The action is a functional with units of J · s given by
S =
∫Ldt
where L is the Lagrangian, which in classical mechanics usually a function of
some generalised coordinates qi and their first derivatives with respect to time, qi.
When a Lagrangian system on a Riemannian manifold is the difference between
kinetic and potential energy of a system it is called natural and corresponds to a
mechanical system [50]. However, the Lagrangian is not necessarily a physically
meaningful quantity; it is not restricted by demands on measurability or gauge
invariance, for instance.
Hamilton’s Principle states that the path followed by the system will be the
one which extremises the action. The equations of motion then come from the
requirement that the Action be an extremum value for the path taken between
two points t1 and t2 in time. In functional calculus, this is equivalent to saying
that the variation of the action must be 0;
δS =
∫ t2
t1
δLdt =
∫ t2
t1
(∂L
∂qδq +
∂L
∂qδq
)dt = 0. (2.1)
A variation with respect to time is not explicitly included since the two points
t1 and t2 are fixed. The variations in the path are required to be 0 at the end
points, so that any path taken must at least start and end at t1 and t2 respectively,
which means that δq(t1) = δq(t2) = 0.
32
2.1 Lagrangian Mechanics
Integrating the second term in Eqn 2.1 by parts and using δq = dδq/dt gives
∫ t2
t1
(∂L
∂q− d
dt
(∂L
∂q
))δqdt = 0
and since this must hold for arbitrary variations δq, it must be true that
d
dt
(∂L
∂q
)=∂L
∂q. (2.2)
This is the Euler-Lagrange equation, but of course for n generalised coordi-
nates we have n equations. The Euler-Lagrange equations are in covariant form,
that is, they have the same functional form under any invertible transformation to
a new set of generalised coordinates q(q, q) and ˙q(q, q). More generally, the Euler-
Lagrange equations come from finding the stationary points of S with respect to
a functional derivative
δS
δq= 0,
completely analogous to the way in which stationary points of functions are found.
A specific example will serve to better illustrate the usefulness of the Euler-
Lagrange equations. Consider the Lagrangian for a simple one-dimensional par-
ticle with mass m in a field,
L(x, x) =1
2mx2 − V (x),
where V is the potential energy of the field which is assumed to not change with
time. The Euler-Lagrange equation 2.2 gives
d
dtmx = −∂V
∂x,
which is instantly recognisable as Newton’s second law, F = ma. In fact, the
most well-known physical equations can be found via this procedure given an
33
2.1 Lagrangian Mechanics
appropriate Lagrangian. The Lagrangian density
L = − 1
4µ0
FαβFαβ − JαAα
yields Maxwell’s inhomogeneous equations in the form
F µα,α = −µ0J
µ.
The relativistic Lorentz force,
d
dtmvγ = q(E + v ×B),
is found via the Lagrangian
L = −mc2γ−1 − qφ+ qv ·A
(see §2.4 for this derivation). The Klein-Gordon equation (relativistic Schrodinger
equation), (∂α∂
α +m2c2
~2
)ψ = 0,
comes from the Lagrangian
L = −mc2ψψ − ~2
mψ,αψ
,α.
The Dirac Equation, which describes the wavefunction of spin-1/2 particles,
(iγα∂α −
mc
~
)ψ = 0,
can be found from the Lagrangian
L = −mc2ψψ + i~cψγα∂αψ.
34
2.2 Canonical Derivation of the Maxwell Energy-Momentum Tensor
where γα in the Dirac equation and its Lagrangian represents the four gamma-
matrices [51], not the relativistic Lorentz factor used in the Lorentz force expres-
sion. The complex conjugate of ψ is represented by ψ.
The power of Hamilton’s principle should be clear; the equations of motion
for physical systems can be derived without a priori knowledge of them, provided
that the correct form of the Lagrangian is known. It was already mentioned that
the Lagrangian is often the difference between the kinetic and potential energy of
the system in classical mechanics, and this fact can be used as a guiding principle
in constructing more complicated Lagrangians for new systems.
2.2 Canonical Derivation of the Maxwell Energy-
Momentum Tensor
Recall Eqn (2.2) which was derived from the Principle of Least Action. Using
this standard form of the Euler-Lagrange equations for a Lagrangian L dependent
on some generalised coordinates q and their first derivatives with respect to time
only. Of course, in relativistic theories, the time coordinate is on the same footing
as the space coordinates and the Euler-Lagrange equation is usually expressed as
d
dxµ
(∂L
qi,µ
)=∂L
∂qi.
The total derivative of the Lagrangian with respect to the spacetime coordi-
nates can then be expressed as follows:
dL
dxµ=∂L
∂qiqi,µ +
∂L
∂qi,νqi,νµ +
∂L
∂xµ
=d
dxν
(∂L
∂qi,ν
)qi,µ +
∂L
∂qi,νqi,νµ +
∂L
∂xµ
=d
dxν
(∂L
∂qi,νqi,µ
)+∂L
∂xµ
=⇒ − ∂L
∂xµ=
d
dxν
(∂L
∂qi,νqi,µ − δνµL
)
35
2.2 Canonical Derivation of the Maxwell Energy-Momentum Tensor
Note that in the preceding derivation, the indices following a comma represent
a total derivative of the generalised coordinates. This was done for notational
simplicity but it is generally clear from the context of a particular problem as
to whether a total or partial derivative is called for. In this way, the partial
derivative of the Lagrangian with respect to the spacetime coordinates (usually
the gradient of the potential fields which is the force) is expressed in terms of
the four-divergence of a tensor which shall be referred to here as the canonical
energy-momentum tensor T νµ
T νµ =
∂L
∂qi,νqi,µ − δνµL. (2.3)
The Maxwell stress tensor is derived from the Lagrangian
L =ε02E2 − 1
2µ0
B2
using the canonical definition of the energy-momentum tensor T nm with respect
to the generalised coordinates φ and Ai such that
T nm =
∂L
∂φ,nφ,m +
∂L
∂Ai,nAi,m − δnmL. (2.4)
This gives
T nm = ε0φ
,nφ,m + ε0∂An
∂tφ,m −
1
µ0
εinqεilmAm,lAq,m + cε0δ
n0φ
,qAq,m + cε0δn0
∂Aq
∂tAq,m
− δnmL
= −ε0Enφ,m −1
µ0
εinqBiAq,m − cε0δn0EqAq,m − δnmL
Taking the divergence and rearranging some terms,
36
2.2 Canonical Derivation of the Maxwell Energy-Momentum Tensor
T nm,n =
∂
∂xn(ε0E
nEm − δnmL) + ε0EnAm,n∂t− 1
µ0
εinqBi,nAq,m −1
µ0
εinqBiAq,mn
− ∂ε0EnAn,m∂t
=∂
∂xn(ε0E
nEm − δnmL) + ε0EnAm,n∂t
+ ε0∂En
∂tAn,m −
1
µ0
BnBn,m −
∂ε0EnAn,m∂t
=∂
∂xn(ε0E
nEm − δnmL) + ε0En∂Am,n − An,m
∂t+
1
µ0
Bn(B ,nm −Bn
,m)
− 1
µ0
BnB,nm
=∂
∂xn(ε0E
nEm −1
µ0
BnBm − δnmL) + ε0En∂Am,n − An,m
∂t− 1
µ0
Bn(B ,nm −Bn
,m)
+2
µ0
Bn(B ,nm −Bn
,m)
=∂
∂xn
(ε0E
nEm +1
µ0
BnBm − δnm(ε02E2 +
1
2µ0
B2
))+ ε0E
n∂Am,n − An,m∂t
− 1
µ0
Bn(B ,nm −Bn
,m)
In vector notation, this becomes the familiar expression for the force in terms
of the divergence of the Maxwell stress tensor U, plus the time derivative of the
Poynting vector S,
f = ∇ ·U− 1
c2∂S
∂t,
such that
f = ∇ ·(ε0E⊗ E +
1
µ0
B⊗B− 1
2I
(ε0E
2 +1
µ0
B2
))− ε0
∂E×B
∂t(2.5)
where I is the 3×3 identity matrix. Therefore, it is not just the equations of mo-
tion that can be found in a Lagrangian framework, but also the conservation laws
of the system. This idea is expressed more generally in Noether’s theorem, which
states that for every ignorable generalised coordinate in the Lagrangian, there is
a corresponding conserved quantity [52]. For instance, a Lagrangian system that
does not depend on time is one that conserves energy. A Lagrangian system that
is invariant under any translation of the coordinates conserves momentum. Ro-
37
2.3 Lagrange Multipliers versus Constrained Variations
tational invariance implies conservation of angular momentum, and so on. The
power of Noether’s theorem is that it can be applied to any set of generalised
coordinates to find the corresponding conserved quantities in the system, which
are often not obvious from the nature of the problem.
2.3 Lagrange Multipliers versus Constrained Vari-
ations
The above description of Hamilton’s principle is a special case where the coor-
dinates q and q are varied independently of each other (and no second order or
higher derivatives of q appear in the Lagrangian). In many problems of physical
interest, there are constraints that must be taken into account. A constraint that
depends only on the generalised coordinates (and possibly time) is referred to as
a holonomic constraint. A constraint that depends on the velocities is nonholo-
nomic or anholonomic. Such constraints are path-dependent and non-integrable.
There are two equivalent ways in which constraints can be introduced into
Hamilton’s principle. The first method of introducing a constraint on the gener-
alised coordinates and/or velocities is via a Lagrange multiplier. Lagrange multi-
pliers are familiar to anyone with a basic knowledge of maximising and minimis-
ing problems in calculus. As an example, consider the free space electromagnetic
Lagrangian density,
L = − 1
2µ0
B2 +ε02E2.
Attempting to retrieve Maxwell’s equations from Hamilton’s principle would
yield trivial solutions if E and B were varied independently. Of course, in elec-
tromagnetism the electric and magnetic fields are constrained by their expression
38
2.3 Lagrange Multipliers versus Constrained Variations
in terms of the electric and magnetic potentials φ and A such that
E = −∇φ− ∂A
∂t,
B = ∇×A.
If the equations of motion,
∇ · E = 0,
∇×B− 1
c2∂E
∂t= 0,
are known a priori, then they can be introduced to the Lagrangian with the aid
of a Lagrange multiplier. Hamilton’s principle would then yield the expressions
for E and B in terms of the potentials. This can be seen with the following
Lagrangian dependent on the electric and magnetic fields, as well as four new
variables in the form of the Lagrange multipliers λi and α such that
L = − 1
2µ0
B2 +ε02E2 + λ ·
(1
c2∂E
∂t−∇×B
)+ α∇ · E.
The six Euler Lagrange equations (technically ten - three components of E
and B each as well as four for the Lagrange multipliers which simply return the
constraints) then give
∂L
∂E− ∂
∂t
∂L
∂ ∂E∂t
− ∂
∂xi∂L
∂ ∂E∂xi
= 0 =⇒ E =1
ε0∇α + µ0
∂λ
∂t,
∂L
∂B− ∂
∂t
∂L
∂ ∂B∂t
− ∂
∂xi∂L
∂ ∂B∂xi
= 0 =⇒ B = −µ0∇× λ.
Therefore, Hamilton’s principle returns the constraints on E and B in terms
of the potentials where the Lagrange multipliers are revealed as
λ = − 1
µ0
A and α = −ε0φ.
39
2.3 Lagrange Multipliers versus Constrained Variations
Alternatively, if the constraints on E and B in terms of the potentials were
known a priori, then the equations of motion could be derived from the La-
grangian
L = − 1
2µ0
B2 +ε02E2 + λ · (B−∇×A) + α ·
(E +∇φ+
∂A
∂t
)
where there are now six new variables corresponding to the three components of
the vectors λ and α.
The second method of dealing with constraints is the method of constrained
variations where the Lagrangian is unchanged, but the variations of the coordi-
nates in Hamilton’s principle are done with respect to the constraints. Continuing
the use of electromagnetism as an example, assume that the coupling between E
and B is known so that the variations δE and δB are connected via the potentials
φ and A such that
δE = −∇δφ− ∂δA
∂t,
δB = ∇× δA.
The Lagrangian itself remains unchanged as
L = − 1
2µ0
B2 +ε02E2,
but the application of Hamilton’s principle now looks like
δS =
∫ (∂L
∂E· δE +
∂L
∂B· δB
)d3xdt
=
∫ (∂L
∂E·(−∇δφ− ∂δA
∂t
)+∂L
∂B· ∇ × δA
)d3xdt
=
∫− ∂
∂t
(∂L
∂E· δA
)−∇ ·
(∂L
∂B× δA +
∂L
∂Eδφ
)− δφ∇ · ∂L
∂E
+ δA ·(∂
∂t
∂L
∂E+∇× ∂L
∂B
)d3xdt = 0.
40
2.3 Lagrange Multipliers versus Constrained Variations
The independent variations of φ and A then give Maxwell’s equations,
δφ : ∇ · E = 0,
δA :1
c2∂E
∂t−∇×B = 0.
Interestingly, this constrained variational principle has also yielded some extra
terms in the integrand in the form of a four-divergence (partial derivative of time
plus divergence in space). This suggest that Noether’s theorem can be applied to
also retrieve information about the conserved quantities in the system. Energy
conservation is given by considering infinitesimal time displacements δt such that,
in this case,
δA =∂A
∂tδt and δφ =
∂φ
∂tδt.
The remaining terms in the varied action integrand (now varied with respect
to δt),
− ∂
∂t
(∂L
∂E· ∂A
∂t
)−∇ ·
(∂L
∂B× ∂A
∂t+∂L
∂E
∂φ
∂t
)= 0,
give
1
2
∂
∂t
(−ε0E2 +
1
µ0
B2
)= 0.
In free space, this is the energy conservation law
1
2
∂
∂t
(−ε0E2 +
1
µ0
B2
)= −1
2
∂
∂t
(ε0E
2 +1
µ0
B2
)−∇ · S = 0.
Considering infinitesimal spatial displacements δx such that
δA = (δx · ∇)A and δφ = (δx · ∇)φ
gives the conservation law with respect to translational invariance, which is the
momentum conservation law for the electromagnetic field in free space,
∇ ·U =1
c2∂S
∂t,
41
2.4 Canonical Derivation of the Lorentz Force
where U is the Maxwell stress tensor derived previously in §2.2. The same proce-
dure shown above can be repeated for constrained variations in terms of Maxwell’s
equations, which would yield expressions for the electric and magnetic field in
terms of the potentials.
In this section, the problem of constraints on a Lagrangian system has been
dealt with using the well-known example of the electromagnetic field equations.
In both the Lagrange multiplier and constrained variation methods, it was shown
that knowledge of one set of constraints - either the form of the equations of
motion or the relationship of the vector fields with the potentials - yielded in-
formation about the other set of constraints. While the results are the same for
either method, it is this author’s opinion that the constrained variation approach
is slightly superior in that the conservation laws are also made immediately made
apparent by Hamilton’s principle. In the case of Lagrange multipliers, slightly
more working would be required to derive the energy-momentum tensor and ap-
ply Noether’s theorem. However, it is also worth noting that the constrained
variation approach was slightly more algebraically complex in this particular ex-
ample.
Both the constrained variation method and Lagrange multipliers are used by
authors whose work is reviewed in Chapter 3.
2.4 Canonical Derivation of the Lorentz Force
The Lorentz force,
f = q(E + v ×B),
can also be derived using a Lagrangian formalism (note that some authors refer
to just the v×B term as the ‘Lorentz force’ but this thesis uses the name for the
total force). The total Lagrangian for a charged particle in an electromagnetic
field is
L(Aµ, Aµ,ν , ξi, ξi) = − 1
4µ0
FαβFαβ − JαAα +mc2(γ − 1) (2.6)
42
2.4 Canonical Derivation of the Lorentz Force
where Jα = ρ0ξα is the four current and ρ0 is the rest frame charge density. Note
the use of the label ξi for the particle coordinates, instead of the usual xi. This
is done to highlight that the dynamic variables ξi refer to the particle position, a
very different concept to the fixed field coordinates xµ on which the generalised
coordinates Aµ depend. This is a subtle point that is often not addressed by
other authors, but the situation is analogous to the difference between Eulerian
and Lagrangian fluid coordinates discussed in §1.3.1. Both the four velocity and
Lorentz factor γ can be expressed in terms of just the three coordinates ξi.
The term on the left in (2.6) is the pure field term, while the term on the right
is the pure particle term (kinetic energy of the particle). The middle term gives
the interaction between the particle and the field. Note that the kinetic energy
of the particle is given by the total relativistic energy E minus the rest energy
mc2 where
E =√m2c4 +m2v2γ2c2 = mc2γ
and v = ξ is the particle velocity. The reason for the addition of the constant
rest energy to the Lagrangian will become clear in a moment.
The Euler-Lagrange equations with respect to the generalised coordinates Aµ
in (2.6) give the electromagnetic field equations of motion (discussed at length
in §2.3), while the Euler-Lagrange equations with respect to ξi give the particle
equations of motion. To derive the Lorentz force, only the particle and interaction
terms depend on ξi, so only these two terms are required such that
L(ξi, ξi) = −JαAα +mc2(γ − 1). (2.7)
For the Lagrangian in (2.7), the action would be given with respect to proper
time τ such that
S =
∫Ld3xdτ,
but a Lagrangian expressed in vector notation can be found given that γdτ = dt
43
2.4 Canonical Derivation of the Lorentz Force
so that
S =
∫(−γ−1JαAα −mc2(γ−1 − 1))γd3xdτ =
∫(−qφ+ qv ·A−mc2γ−1)d3xdt,
where the constant rest energy has now been dropped from the left hand side,
and a Lagrangian equivalent to (2.7) is therefore
L = −qφ+ qv ·A−mc2γ−1. (2.8)
Plugging the Lagrangian from (2.8) into the Euler-Lagrange equations gives
∂L
∂ξi=
d
dt
(∂L
∂vi
)−qφ,i + qvjAj,i = q
dAidt
+mdviγ
dt
−qφ,i − q∂Ai∂t
+ qvjAj,i − qvjAi,j = mdviγ
dt
Note that ξ has now been identified with xi in the derivatives of the potentials
since the particle position within the field must be considered in the interaction
term, analogous to the way in which an Eulerian velocity vector field at a certain
point in space is identified with the Lagrangian fluid particle velocity at that
point in time. Using the definition of the electric and magnetic fields where
E = −∇φ− ∂A
∂tand B = ∇×A,
the Lorentz force can be expressed in vector notation now as
dmvγ
dt= q(E + v ×B).
Since the Lorentz force is derived only from the Lagrangian terms correspond-
ing to the kinetic energy of the particle and the interaction term, it is clear that
44
2.5 Hamiltonian Mechanics
adding non-standard field terms to the Lagrangian will have no effect on the
Lorentz force. This will be crucial in §4.3 where the Podolsky Lagrangian - a
non-standard Lagrangian for electrodynamics - is applied to the case of laser-
plasma interaction.
2.5 Hamiltonian Mechanics
Hamiltonian mechanics was developed as an alternative to the Lagrangian formu-
lation of classical mechanics. While in classical mechanics the Lagrangian is the
difference in kinetic and potential energy of a system, the Hamiltonian is their
sum, equal to the total energy of the system. The Hamiltonian description is
usually not easier to work with or develop than the Lagrangian description, but
it has many edifying qualities not found in Lagrangian mechanics. A Hamiltonian
gives the equations of motion for 2n variables in terms of first order differential
equations, as opposed to the Euler-Lagrange equations which are second order
differential equations for n generalised coordinates. The true value of Hamilto-
nian mechanics in modern physics lies in its applicability to quantum mechanics
where ‘canonical quantization’ of a system converts the total energy into a Hamil-
tonian operator that generates time evolution of the system. For instance, the
time-dependent Schrodinger equation is expressed as
i~∂ψ
∂t= Hψ
where ψ is the wavefunction of the system and H the Hamiltonian operator,
found by canonical quantization of its coordinates and momenta. Without a
well-defined Hamiltonian, a quantum description of a system is not possible.
Textbook descriptions of Hamiltonian mechanics usually begin with a La-
grangian function before proceeding to the Hamiltonian H(p, q, t) via a Legendre
transformation,
H(p, q, t) = piqi − L(q, q, t) (2.9)
45
2.5 Hamiltonian Mechanics
where the canonical momentum p is
pi =∂L
∂qi. (2.10)
The derivation of Hamilton’s equations is elementary and found in any good
textbook on classical mechanics, e.g. [42]. Consider the differential of the Hamil-
tonian,
dH(q, p, t) =∂H
∂qdq +
∂H
∂pdp+
∂H
∂tdt
=∂(pq − L)
∂qdq +
∂(pq − L)
∂pdp+
∂(pq − L)
∂tdt
= − d
dt
(∂L
∂q
)dq + qdp− ∂L
∂tdt (using 2.2)
= −pdq + qdp− ∂L
∂tdt.
Equating coefficients of the differentials dq, dp and dt in the first and last lines
above gives Hamilton’s equations:
∂H
∂q= −p, ∂H
∂p= q,
∂H
∂t= −∂L
∂t. (2.11)
Given any arbitrary density function in phase space f(q, p, t), its total time
derivative is
df
dt=∂f
∂t+∂f
∂qiqi +
∂f
∂pipi.
If Hamilton’s equations are used to replace q and p, then
df
dt=∂f
∂t+∂f
∂qi
∂H
∂pi− ∂f
∂pi
∂H
∂qi=∂f
∂t+ {f,H}
where the Poisson bracket is
{f, g} :=∂f
∂qi
∂g
∂pi− ∂g
∂qi
∂f
∂pi. (2.12)
46
2.5 Hamiltonian Mechanics
If the density function f obeys Liouville’s theorem then df/dt = 0 [see 20,
Ch. 1, §3] and the time evolution of f is given by
∂f
∂t= −{f,H}. (2.13)
Indeed, it is clear that Hamilton’s equations for q and p are also expressible
in terms of the Poisson bracket
q = {q,H}, p = {p,H}, (2.14)
and this is the essential feature of a Hamiltonian theory - the time evolution of
any quantity in phase space is said to be generated by the Hamiltonian via the
Poisson bracket.
The significance of the Poisson bracket extends into many branches of math-
ematics and physics. In quantum mechanics, a Hamiltonian theory has the same
essential features as in classical mechanics except that the Poisson bracket goes
over to a simple commutator with the property that the position and momentum
operators do not commute,
[x, p] =
[x,−i~ ∂
∂x
]= i~.
A transformation to a new set of coordinates in phase space is said to be
canonical if it preserves the form of Hamilton’s equations. It is not necessary
to preserve the functional form of the Hamiltonian itself. This means that a
transformation of coordinates p → P and q → Q will result in a transformed
Hamiltonian H → K, but Hamilton’s equations stay the same with respect to
the new Hamiltonian K,
Q = {Q,K}, P = {P,K}.
Since a physicists goal is usually to find the essential conserved quantities or
47
2.5 Hamiltonian Mechanics
symmetries for any system, it is convenient to find a canonical transformation
to new coordinates Q and P such that Q = P = 0. The easiest way to do this
is to canonically transform to a new Hamiltonian K such that K = 0, making
the equations of motion trivial but instantly revealing the relevant conserved
quantities as the new coordinates and their conjugate momenta.
Consider that any canonical transformation must leave the variation of the
action the same:
δS = δ
∫(pq −H)dt = δ
∫(PQ−K)dt = 0
although this does not mean the integrands themselves are exactly the same. The
variation of the action is unaltered up to the addition of any total time derivative
of a (at least) twice differentiable function of the generalised coordinates, F (q, t).
To see this, consider that
δS = δ
∫ b
a
(L(q, q, t) +
dF (q, t)
dt
)dt
=
∫ b
a
δLdt+∂F
∂qδq
∣∣∣∣ba
,
and the variation of q vanishes at the endpoints a and b, so the addition of dF/dt
to the action integrand has no effect on the variation of the action. Thus, a
canonically transformed action integrand must be related in general by
pq −H = PQ−K +dF (q,Q, t)
dt
(a scale transformation may also relate these expressions, but it is a trivial case).
This gives (p− ∂F
∂q
)q −H =
(P +
∂F
∂Q
)Q−K +
∂F
∂t
and since the coordinates q and Q are independent of each other, their coefficients
48
2.5 Hamiltonian Mechanics
must vanish [42], ultimately yielding the equations
p =∂F
∂q, P = −∂F
∂Q, and K = H +
∂F
∂t.
Since a new Hamiltonian K = 0 is desirable to simplify the equations of
motion, this leaves
H +∂F
∂t= 0
and the function F takes on a very special form, since its total time derivative is
dF
dt=∂F
∂t+∂F
∂qq +
∂F
= pq −H (Q = 0 when K = 0),
which reveals F in this case to be the action itself (up to a constant).
Eqns (2.14) can be given by some other ‘noncanonical’ bracket and still indi-
cate a Hamiltonian theory, provided that the bracket is bilinear, antisymmetric
and satisfies the Jacobi identity [53],
{f, {g, h}}+ {h, {f, g}}+ {g, {h, f}} = 0.
Interestingly, a proof due to Darboux states that such a noncanonical bracket
can always be transformed back into the canonical Poisson bracket, at least locally
in some neighbourhood around a point in phase space [54].
This fact has been exploited by some [55, 56, 57, 57, 58, 59, 60, 61], includ-
ing this author [62], in deriving noncanonical Poisson brackets or noncanonical
coordinates to yield a Hamiltonian description of systems that were thought to
have no Hamiltonian structure. This is extremely useful, since many systems of
real physical interest are described in terms of variables that defy all attempts
to transform them into canonical coordinates. The Hamiltonian may however be
defined independent of a Lagrangian and in terms of coordinates that are not nec-
essarily canonical, provided the noncanonical Poisson bracket satisfies the three
49
2.5 Hamiltonian Mechanics
properties mentioned above. For an excellent review on noncanonical methods
applied to fluids (and Hamiltonian mechanics in general), see [63].
Note that the definition of the Hamiltonian in Eqn (2.9) requires a concave
Lagrangian function and that Eqn (3.7) be invertible to q (otherwise H would
not be expressible in terms of just p, q). Dirac developed a more general theory,
known as Dirac constraint theory [5], that sidesteps these requirements. Given
a number of constraints φk(p, q) = 0 arising from problems with Eqns (2.9), or
merely from other constraints one might choose to impose on the system, the
equations of motion become
dqidt≈ {qi, H}+ uk{qi, φk},
where the symbol ‘≈’ represents weak equality, indicating that strong equality
follows by applying the constraints only after the Poisson brackets have been eval-
uated. The coefficients uk are determined by the requirement that the constraints
are constant in time;
dφidt≈ {φi, H}+ uk{φi, φk} ≈ 0.
Provided that the Lagrangian from which the Hamiltonian was derived is
consistent, there are only two relevant outcomes of evaluating the above equation.
Either a new secondary constraint is found (a function expressed just in terms
of the canonical variables), or a condition is found that the coefficient uk must
satisfy. If a secondary constraint is found, then this process must be repeated
again, and again for any further constraints found, until all that is left is a number
of constraints and conditions on their corresponding coefficients.
The total Hamiltonian HT is written as the ‘naive’ Hamiltonian H, con-
structed from a Legendre transformation of the Lagrangian, plus the constraints
50
2.5 Hamiltonian Mechanics
φk(q, p) multiplied by their corresponding coefficients uk so that
HT = H + ukφk.
The equations of motion for the canonical coordinates are then correct with
respect to the time evolution generated by HT ;
dq
dt≈ {q,HT}
where the constraints are applied after evaluating the Poisson bracket. The use of
Hamiltonian mechanics in fluid and plasma physics will be reviewed in Chapter
3.
51
3
Lagrangian and Hamiltonian
Formulations of Laser-Plasma
Interaction
The focus of this thesis is on variational principles applied to laser-plasma inter-
action using either a fluid model or a kinetic model. This chapter will serve as
a review of all the relevant literature related to variational principles applied to
both fluid and kinetic models. While some of the literature does not deal specifi-
cally with plasmas, the underlying mathematical principles are the same for any
fluid or kinetic model, give or take certain assumptions that may be relevant
only to a plasma. All of this literature has a common thread in that they at-
tempt to describe the physics of fluids or gases using Lagrangian or Hamiltonian
formalisms.
Variational principles in fluid dynamics were used as early as 1929 by Bateman
[64], but the truly influential papers on the subject were due to the likes of Taub
[65, 66], Davydov [67], Herivel [68] and Eckart [69].
52
3.1 The Boltzmann-Vlasov Distribution
3.1 The Boltzmann-Vlasov Distribution
A canonical description specifically tailored for plasmas was first considered by
Low [44] who sought to generalise Taub’s hydrodynamic method [65] to a plasma
described by a Boltzmann-Vlasov equation. Low considered an initial Boltzmann-
Vlasov distribution (at time t = 0) given by f(x0,v0) and let the particle position
at any time t, x(x0,v0, t), be a solution to the equation of motion that returns
the initial coordinates and velocities when t = 0 such that
x(x0,v0, 0) = x0 and v(x0,v0, 0) = v0.
Under a change of coordinates from x → x0 and v → v0, which has unit
Jacobian, the Boltzmann-Vlasov distribution can be transformed such that
f(x,v, t)→ f(x0,v0).
The total plasma Lagrangian L is then given by the integral of a Lagrangian
density L for matter and interaction with respect to the distribution function
over all of phase space, plus the total electromagnetic energy;
L =
∫ ∫f(x0,v0)Ldx0dv0 +
∫ (ε02E2 − 1
2µ0
B2
)dx.
If the Lagrangian density is the standard one for a particle in an electromag-
netic field,
L =1
2m
(∂x
∂t
)2
− qφ+ q∂x
∂t·A,
then Low found that Hamilton’s Principle,
δS =
∫δLdt = 0,
where the quantities x, φ and A are independently varied yields the Lorentz force
53
3.2 Ideal Fluids
and Maxwell’s inhomogeneous equations:
m∂2x
∂t2= q(E + v ×B);
∇ · E =q
ε0
∫f(x,v, t)dv;
∇×B = µ0q
∫f(x,v, t)dv +
1
c2∂E
∂t.
3.2 Ideal Fluids
An ideal fluid description using Hamilton’s principle was given by Herivel [68]
who was unsatisfied by the earlier efforts of Lichtenstein [70] and Taub [65].
Lichtenstein had assumed no variation in temperature in applying Hamilton’s
principle, while Taub had defined temperature as the time derivative of another
(physically meaningless) function and then applied the constraint of constant
entropy after the application of Hamilton’s principle. These approaches seemed
ad hoc to Herivel, who considered the true power of Hamilton’s principle to lie
in its ability to derive equations of motion, rather than assume them, in the
process of describing a system. Herivel proposed that moving from a discrete to
continuous (fluid) description requires changing the form of the Lagrangian by
subtracting an additional term, the internal energy U(n, S) of the fluid, so that
L = T − V − U where T is the kinetic energy of the fluid and V the potential
energy.
In §1.3.1, two equivalent formulations of fluid mechanics were reviewed using
either Eulerian or Lagrangian coordinates. Herivel considered both cases and
used essentially the same approach that Low [44] later applied to the Maxwell-
Boltzmann description of a plasma by transforming the volume integral in the
action into one depending on the initial Lagrangian coordinates a, rather than
54
3.2 Ideal Fluids
the usual Lagrangian coordinates ξ(a, t) so that
S =
∫ ∫ (1
2nξ2 − n(V + U)
)Jdadt
where
J =∂ξ
∂a.
In this way, the action integral uses a volume element comoving with the fluid,
and all that remains is to apply the constraint
n(ξ, t) =n(a)
J(3.1)
(a form of the continuity equation in Lagrangian coordinates) and the isentropic
constraint (see §1.3.3). In Lagrangian coordinates, constant entropy is expressed
as (∂S
∂t
)a
= 0. (3.2)
The above method of transforming coordinates and functions is ubiquitous
in the literature discussing fluid mechanics using Lagrangian coordinates [see
61, 68, 69, 71, 72, 73, 74]. Both (3.1) and (3.2) can be added to the action
integral using Lagrange multipliers α and β such that
S =
∫ ∫ (1
2nξ2 − n(V + U)− α(n(ξ, t)J − n(a))− β
(∂S
∂t
)a
)dadt.
Applying Hamilton’s principle where ξ, n and S are independently varied
(variation with respect to α and β simply returns the constraints), Herivel ulti-
mately found the set of equations:
δn : α =1
2ξ2 − V − U − P
n
δS :∂β
∂t= nT
δξ :∂2ξ
∂t2= −∇V − 1
n∇P
55
3.2 Ideal Fluids
The last equation found by varying ξ is a form of Euler’s equation (1.21) that
includes the gradient of any other potential field that may affect the fluid (gravity,
for instance). The same basic procedure was used relatively recently by Antoniou
& Pronko [74] to give a Hamiltonian description of a plasma.
Herivel achieved the same results in Eulerian coordinates. In this case, the
volume integral in the action is with respect to the Eulerian coordinates x and
the velocity is the Eulerian velocity v(x, t). The functions n and S are also now
expressed in Eulerian coordinates. As such, the constraints must be modified for
the new coordinates, where the continuity equation is the usual
∂n
∂t+∇ · (nv) = 0
and constant entropy flux density is given by
ndS
dt= 0
where d/dt is now of course the material derivative. Inserting these constraints
into the action using Lagrange multipliers α and β gives
S =
∫ ∫ (1
2nv2 − n(V + U)− α
(∂n
∂t+∇ · (nv)
)− βndS
dt
)dxdt.
In the Lagrangian case, the third variable apart from n and S that was to be
independently varied was the fluid coordinate ξ. Now in the Eulerian case, the
velocity will be varied independently in applying Hamilton’s principle and the set
of equations Herivel found from these variations (after some working) is
δn :dv
dt= −∇V − 1
n∇P
δS :∂β
∂t+ v · ∇β = 0
δv : v = −∇α + β∇S
56
3.3 Clebsch Potential Representation
Euler’s equation has again been recovered by an application of Hamilton’s
principle, but this time with respect to the Eulerian coordinates. Herivel noticed
that the final equation from δv represents a decomposition of the vector field
with respect to three scalar functions, which is a special case of something called
the Clebsch potential representation that will be discussed in §3.3). This was the
weakness of Herivel’s Eulerian approach; the velocity is expressed only in terms
of ‘unphysical’ potentials and is not in fact fully general [71]. An alternative to
the above method of varying the Eulerian velocity will be discussed in §3.4.
3.3 Clebsch Potential Representation
Eckart [69] generalised Herivel’s approach to the case of a compressible fluid, but
it was Lin [75] who noticed that an additional constraint introduced to the action
would make Herivel’s Eulerian velocity expression fully general. The additional
term was of the form
ndλidt,
introduced as a constraint to the action integral with the Lagrange multiplier γi.
With this extra constraint, the variation with respect to the Eulerian velocity
yielded the expression
v = −∇α + β∇S + γi∇λi.
The velocity was now a fully general expression at the cost of introducing six
new scalar functions with no intuitive physical interpretation. Seliger & Whitham
[76] showed that Lin’s generalisation could be simplified by pointing out the
connection with a result of Clebsch [77]. Clebsch had shown that any vector field
could be generally decomposed into an expression in terms of only three scalar
functions such that
v = α∇β +∇ψ. (3.3)
Chen & Sudan [78] used the Clebsch potentials specifically in their research
57
3.3 Clebsch Potential Representation
related to laser-plasma interaction. They considered the canonical relativistic
momentum in an electromagnetic field given by
P = mvγ − qA = α∇β +∇ψ. (3.4)
For the special case of a curl free vector field (∇ × P = 0), the decompo-
sition is simply P = ∇ψ since in general ∇α × ∇β 6= 0. Chen & Sudan were
interested in cold relativistic plasmas. The equations governing cold, relativistic
laser-plasma fluid interactions are Maxwell’s two inhomogeneous equations (as
usual, the homogeneous equations follow from the definition of E and B in terms
of the potentials), the Lorentz force and the continuity equation:
∇×B = −µ0J +1
c2∂E
∂t;
∂p
∂t+ v · ∇p = q(E + v ×B);
∂n
∂t= −∇ · nv.
Note that the Lorentz force was given in terms of the material derivative of
the momentum p, which must be so in the case of fluid dynamics.
The relativistic Lagrangian for laser-plasma interaction selected by Chen &
Sudan was
L = nmc2(1− γ−1) + qnφ− qnv ·A− 1
2µ0
B2 +ε02E2. (3.5)
Given the vector calculus identity,
∇(A ·B) = A · ∇B + B · ∇A + A× (∇×B) + B× (∇×A), (3.6)
58
3.3 Clebsch Potential Representation
the Lorentz force can be expressed in terms of the canonical momentum:
∂P
∂t= −v · ∇mvγ +∇qφ− qv × (∇×A)
= −v · ∇mvγ +∇qφ− v × (∇× (mvγ −P))
= ∇(qφ−mv2γ) +mvγ · ∇v + v × (∇×P) (using 3.6)
= ∇(qφ−mv2γ −mc2γ−1) + v × (∇×P)
= ∇(qφ−mc2γ) + v × (∇×P).
(3.7)
But now, given the ‘Lin constraints’ [75]
dα(x, t)
dt=dβ(x, t)
dt= 0, (3.8)
the Lorentz force can also be expressed in terms of Clebsch potentials:
∂(α∇β +∇ψ)
∂t= ∇(qφ−mc2γ) + v × (∇α×∇β)
∇(α∂β
∂t+∂ψ
∂t
)−∇α∂β
∂t+∇β∂α
∂t= ∇(qφ−mc2γ) + (v · ∇β)∇α− (v · ∇α)∇β
=⇒ α∂β
∂t+∂ψ
∂t= qφ−mc2γ + C (using 3.8)
where C is a constant determined by initial conditions. Since the plasma may be
considered static prior to the arrival of the laser pulse, at time t = 0, γ = 1 and
∂ψ/∂t = α = β = 0 which gives C = mc2 and the Lorentz force is
α∂β
∂t+∂ψ
∂t= qφ−mc2(γ − 1). (3.9)
The Lagrangian itself (3.5) can also be expressed independently of v assuming
59
3.3 Clebsch Potential Representation
that momentum is conserved, i.e., dψ/dt = 0, given that
mc2(1− γ−1)− qv ·A = mc2 +mv2γ −mc2γ − qv ·A
= v · (mvγ − qA)−mc2(γ − 1)
= v · (α∇β +∇ψ)−mc2(γ − 1)
= −α∂β∂t− ∂ψ
∂t−mc2(γ − 1).
Therefore, the Lagrangian 3.5 can be expressed in terms of the Clebsch po-
tentials and electromagnetic potentials as
L = −n(α∂β
∂t+∂ψ
∂t+mc2(γ − 1)− qφ
)− 1
2µ0
B2 +ε02E2. (3.10)
Note that the relativistic factor γ can also be expressed independent of v
given that
γ =1√
1− v2/c2=⇒ γ2v2 = c2(γ2 − 1)
and
P = mvγ − qA =⇒ γ2v2 =(P + qA)2
m2,
which gives
γ =√
1 + (P + qA)2/m2c2.
However, Chen & Sudan argued that prior to a laser pulse hitting the plasma,
the plasma electrons have zero canonical momentum and so the curl of the mo-
mentum is also zero. However, taking the curl of the Eqn (3.7) shows that this
will be true for all time and so the canonical momentum can be considered to be
curl-free in this scenario and expressible in terms of just one potential, ψ. In this
case, the Lagrangian density is just
L = −n(∂ψ
∂t+mc2(γ − 1)− qφ
)− 1
2µ0
B2 +ε02E2 (3.11)
60
3.3 Clebsch Potential Representation
and the Lorentz factor is
γ =√
1 + (∇ψ + qA)2/m2c2.
Now Hamilton’s Principle can be applied to the action
S =
∫Ld3xdt (3.12)
where L is given by 3.11 and the generalised coordinates to be independently
varied are ψ,A, φ, n. However, even though α and β no longer appear in the
action integral, it is worth noting what their variation would yield in the general
case where L is (3.10). Since there are no derivatives of α appearing in 3.10,
independently varying α simply yields the equation
∂L
∂α=∂β
∂t+mc2
∂γ
∂α= 0. (3.13)
The relativistic factor γ is a function of P = α∇β +∇ψ and A, so
∂γ
∂α=
∂
∂α
(1 +
α2(∇β)2 + 2α∇β · ∇ψ + (∇ψ)2 + 2q(α∇β ·A +∇ψ ·A) + q2A2
m2c2
)1/2
=α(∇β)2 +∇β · ∇ψ + q∇β ·A
m2c2γ
= ∇β · mvγ
m2c2γ
= ∇β · v
mc2
Therefore, equation 3.13 returns one of the Lin constraints (the constraint
that β is ‘carried’ along with the fluid flow):
∂β
∂t+mc2
∂γ
∂α=∂β
∂t+ v · ∇β =
dβ
dt= 0.
The reappearance of the velocity in the calculations above is merely a conve-
nience to transit to the final expression.
61
3.3 Clebsch Potential Representation
There are derivatives of β appearing in 3.10, so varying β yields the equation
∂L
∂β− ∂
∂t
∂L
∂β− ∂
∂xi∂L
∂ ∂β∂xi
= 0, (3.14)
which gives
∂α
∂t+ v · ∇α =
dα
dt= 0
since
∂γ
∂∇β=
α
mc2v.
Now to the heart of the matter, for the case where α = β = 0, varying ψ in
(3.12) gives the equation
∂L
∂ψ− ∂
∂t
∂L
∂ψ− ∂
∂xi∂L
∂ ∂ψ∂xi
= 0. (3.15)
Note that the derivatives in equation 3.15 are partial derivatives as it is these
partial derivatives of ψ that are treated as independent variables in the integration
by parts that takes place when finding the action extremal. The distinction
is emphasised here especially to point out that ψ - the time derivative of the
generalised coordinate ψ - is not treated as the material derivative d/dt = ∂/∂t+
v · ∇ for the purposes of applying Hamilton’s principle. Now equation 3.15 gives
the continuity equation,
∂n
∂t+∇ · ( n
mγ(∇ψ + qA)) = 0, (3.16)
Therefore, it has been shown that independently varying α and β in Equation
3.12 and demanding that δS = 0 yields the Lin constraints for β and α respec-
tively, while varying ψ yielded the equation of continuity. This process can be
repeated for the remaining generalised coordinates φ and A. To summarise, the
coordinate variations are matched below with the equation yielded by Hamilton’s
62
3.4 Virtual Fluid Displacement
Principle:
δα :dα
dt= 0,
δβ :dβ
dt= 0,
δψ :∂n
∂t+∇ · ( n
mγ(∇ψ + qA)) = 0,
δφ : ∇ · E =qn
ε0,
δA : ∇×B = µ0qn
mγ(∇ψ + qA) +
1
c2∂E
∂t,
δn :∂ψ
∂t= qφ−mc2(γ − 1).
Chen & Sudan therefore succeeded in applying Hamilton’s principle to the
case of cold, relativistic laser-plasma interaction where the plasma is treated as
a fluid with Eulerian velocity v. Some authors find the decomposition of v in
terms of the Clebsch potentials to be useful as a mathematical tool, but find their
‘unphysicality’ to be a barrier to deeper understanding of a given problem.
3.4 Virtual Fluid Displacement
While Chen & Sudan [78] and Seliger & Whitham [76] considered the Clebsch
potential representation to solve the difficulties of Hamilton’s Principle applied to
a fluid system using Eulerian coordinates, Newcomb had earlier taken a different
approach [79] to a variational formulation of magnetohydrodynamics applied to
the case of an infinitely conductive, non-relativistic plasma. The crux of the
problem was still the difficulties in applying a variational principle to a fluid
theory in terms of Eulerian coordinates (this was the reason that other authors
explored Clebsch potential formulations). Newcomb was able to reproduce certain
magnetohydrodynamic equations in both Eulerian and Lagrangian coordinates
(see §1.3.1) via Hamilton’s principle and demonstrated. Herivel expressed the
pressure P , density n and magnetic field B in Lagrangian coordinates with respect
63
3.4 Virtual Fluid Displacement
to the initial configuration of the system in the usual way (see §3.2)) such that
n =n(a)
J,
P =P (a)
Jκ,
Bi =∂ξi∂aj
Bi(a)
J,
where
J = det
(∂ξ
∂a
)and κ is the adiabatic index (κ = CP/CV - ratio of specific heat at constant
pressure over constant volume). The set of magnetohydrodynamic equations that
Newcomb sought to reproduce (expressed here all in Eulerian coordinates) was
∂n
∂t+∇ · (nv) = 0,
∂P
∂t+ v · ∇P + κP∇ · v = 0,
∂B
∂t−∇× (v ×B) = 0,
∇ ·B = 0,
n
(∂v
∂t+ (v · ∇)v +∇φ
)+∇
(P +
1
2B2
)−B · ∇B = 0.
(3.17)
The first equation above is the usual continuity equation. The second equa-
tion is the adiabatic ideal gas law and the third equation is a combination of
Faraday’s Law with the Lorentz force law where infinite conductivity is assumed
(−E = v × B). The last equation is another way of expressing the Lorentz
force law where the time varying electric field has been neglected in Ampere’s
law in calculating J × B = (∇ × B) × B. This particular set of equations is
non-relativistic and also uses a scalar pressure P , despite the assumption of in-
finite conductivity (usually only assumed for collisionless plasmas with tensorial
64
3.4 Virtual Fluid Displacement
pressure). Newcomb argued that these assumptions are justified in certain as-
trophysical regimes. The particular form of these equations is not particularly
relevant to this thesis, indeed, laser-plasma interaction was not Newcomb’s in-
terest in studying this system of equations, but his work has influenced other
authors who used similar ideas in direct application to laser-plasma physics. It is
the particular way in which Newcomb solved a problem in fluid mechanics using
Hamilton’s principle that is relevant here.
Newcomb guessed the form of the Lagrangian for the system of equations in
(3.17) based on the standard expression as a difference of the kinetic and potential
energy. In Lagrangian coordinates Newcomb’s Lagrangian density was
L = n
(1
2ξ2 − ϕ
)− P
(κ− 1)Jκ−1− 1
2µ0J
∂ξi∂aj
∂ξi
∂akBjBk (3.18)
where ϕ is the gravitational potential energy. Variation of the action dependent
on this Lagrangian density with respect to the Lagrangian coordinates ξ yields
the system of equations in (3.17) once converted back to Eulerian coordinates,
proving that Newcomb’s choice of Lagrangian was the correct one.
The most important ingredient in Newcomb’s work (for the purposes of this
thesis) was his derivation of the variation of the Eulerian fluid velocity, which is of
course necessary for any variational theory attempting to deal with Eulerian fluid
dynamics without the use of Clebsch potentials. Newcomb used the definition of
Eulerian velocity in terms of Lagrangian velocity,
v(x, t)|x=ξ(a,t) = ξ(a, t), (3.19)
together with a definition of virtual fluid displacement z, (variation of Eulerian
fluid coordinates in terms of Lagrangian particle coordinates)
z(x, t)|x=ξ(a,t) = δξ(a, t) (3.20)
65
3.4 Virtual Fluid Displacement
to define a variation in the Eulerian fluid velocity. The variation in the fluid
velocity will be considered while following a particular fluid particle so that the
variation of Eqn (3.19) is done while holding a, the initial position of a Lagrangian
particle, constant so that
δv|x=ξ(a,t) = δξ − (δx · ∇)ξ
=⇒ δv + (z · ∇)v = δξ.
(3.21)
Next, simply take the time derivative of Eqn (3.20) to get
dz
dt
∣∣∣∣x=ξ(a,t)
=∂z
∂t+ (v · ∇)z = δξ. (3.22)
Combining Eqns (3.21) and (3.22) gives
δv =∂z
∂t+ (v · ∇)z− (z · ∇)v, (3.23)
which is the variation of the Eulerian fluid velocity in terms of the virtual fluid
displacement z. The variations in density, scalar pressure and magnetic field can
also be defined in terms of the virtual fluid displacement where
δn = −∇ · (nδξ), (3.24)
δP = −κP (∇ · z)− z · ∇P, (3.25)
and
δB = ∇× (z×B). (3.26)
The Lagrangian from (3.18), now expressed in Eulerian coordinates is
66
3.4 Virtual Fluid Displacement
L = n
(1
2v2 − ϕ
)− P
κ− 1− 1
2µ0
B2.
It was with this expression that Newcomb applied Hamilton’s principle using
Eqns (3.23), (3.24), (3.25) and (3.26) to yield the system of equations set out in
(3.17), showing that his method of variation with respect to the Eulerian virtual
fluid displacement was also correct.
Newcomb’s method of varying the Eulerian fluid velocity was used by Brizard
[80] [81] to specifically study the case of laser-plasma interactions for a cold rel-
ativistic plasma. The Lagrangian considered by Brizard was
L = nmc2(1− γ−1)− q(ne − ni)φ+ qnv ·A− 1
2µ0
B2 +ε02E2, (3.27)
which is simply a relativistic Lagrangian (see §2.4) for two charged fluids, one
corresponding to an electron fluid with density ne and the other to an ion fluid
with density ni. Brizard used Newcomb’s expression for the variation of the
Eulerian fluid velocity and density - Eqns (3.23) and (3.24) - in terms of the
virtual fluid displacement in applying Hamilton’s principle to (3.27).
The variation of (3.27) is therefore given by
δL =∂L
∂nδn+
∂L
∂v· δv +
∂L
∂φδφ+
∂L
∂A· δA +
∂L
∂B· δB +
∂L
∂E· δE
= −∂L∂n∇ · (nδξ) +
∂L
∂v·(∂δξ
∂t+ v · ∇δξ − δξ · ∇v
)+∂L
∂φδφ+
∂L
∂A· δA
+∂L
∂B· (∇× δA) +
∂L
∂E·(−∇δφ− ∂δA
∂t
)= δφ
(∂L
∂φ+∇ · ∂L
∂E
)+ δA ·
(∂L
∂A+∂
∂t
(∂L
∂E
)+∇× ∂L
∂B
)− δξ ·
(∂
∂t
(∂L
∂v
)− n∇∂L
∂n+∇ ·
(∂L
∂v⊗ v
)+∂L
∂v· ∇v
)+∂
∂t
(∂L
∂v· δξ − ∂L
∂E· δA
)+∇ ·
(∂L
∂B× δA− ∂L
∂nnδξ +
∂L
∂v· (v ⊗ δξ)− ∂L
∂Eδφ
)
67
3.4 Virtual Fluid Displacement
Note that the varied Lagrangian is now split into terms multiplied by the arbi-
trary variations of φ,A, ξ, a partial time derivative and a partial space derivatives.
Hamilton’s principle yields the set of equations,
δφ : ∇ · E =qn
ε0,
δA : ∇×B = µ0qnv +1
c2∂E
∂t,
δξ :∂p
∂t+ v · ∇p = q(E + v ×B).
The remaining part of the varied Lagrangian gives the energy conservation
and momentum conservation equations upon application of Noether’s theorem.
For the equations derived above, the field variables were varied independently,
that is their functional form was varied while holding their spacetime dependence
constant. Consider now time translations of the variables φ and A, where the
functional form is held constant while the time dependence is varied:
δA =∂A
∂tδt, δφ =
∂φ
∂tδt and δξ = vδt.
The equation regarding conservation of energy is then given by
∂
∂t
(∂L
∂v· v − ∂L
∂E· ∂A
∂t
)(3.28)
+∇ ·(∂L
∂B× ∂A
∂t− n∂L
∂n⊗ v +
∂L
∂v· (v ⊗ v)− ∂L
∂E
∂φ
∂t
)δt = 0
which is the same law of conservation of energy derived in §2.3, except where a
current is present;
1
2
∂
∂t
(ε0E
2 +1
µ0
B2
)+∇ · S = −qnv · E.
The law of conservation of momentum is found by considering spatial trans-
lations of the variables φ and A where
68
3.5 Relativistic Constraints in Fluid Dynamics
δφ = (δx · ∇)φ, δA = (δx · ∇)A and δξ = δx,
so that Hamilton’s principle gives
∂
∂t
(∂L
∂v· v − ∂L
∂E· ∇A
)δx (3.29)
+∇ ·(∂L
∂B×∇A− ∂L
∂nn⊗ v +
∂L
∂v· (v ⊗ v)− ∂L
∂E∇φ)δx = 0
which is
∇ ·U− 1
c2∂S
∂t=dnv
dt.
The equations governing relativistic cold laser-plasma interaction can there-
fore be derived using Eulerian coordinates. The Clebsch potential formalism
relies on the ‘unphysical’ potential functions α, β and ψ and the emergence of
the equations from a variational principle is not intuitive. In contrast to this, the
Eulerian formulation elaborated upon by Brizard gives the Lorentz force law from
the variation of the virtual fluid displacement analogous to the way Newton’s law
of motion can be found from varying the physical coordinates of a particle in
classical mechanics.
Brizard et al. [82] also used this general method of variation in considering
neutrino-plasma interactions using a fluid model, although a discussion of neutri-
nos is beyond the scope of this thesis. The subject of nonlinear plasma dynamics,
treated as a fluid, was also explored in detail by Zakharov et al. [61, 83] using a
Hamiltonian theory that did not rely on a Lagrangian description to begin with.
3.5 Relativistic Constraints in Fluid Dynamics
In special relativity, the proper time and four-velocity are no longer independent
of each other but are constrained by the relationship
dξαdξα = c2dτ 2.
69
3.5 Relativistic Constraints in Fluid Dynamics
Therefore, a variation of the infinitesimal proper time is
δdτ =1
cδ(√dξαdξα) =
1
c2dξαdτ
dδξα (3.30)
which implies a variation in the four-velocity
δ
(dξµdτ
)=dδξµdτ− dξµ
(dτ)2δdτ =
dδξµdτ− 1
c2dξµdτ
dξαdτ
dδξα
dτ. (3.31)
These constraints were addressed long ago by Infeld [84] and Kalman [85].
Cavalleri and Spinelli [86, 87, 88] attempted to take into account curvilinear co-
ordinates in General Relativity or relativistic compressible fluids using the above
variation, research that was repeated by Brown [89] many years later, perhaps
more rigorously. Kalman noted [85] that, given the constraint 3.31 (which also
implies uαuα = c2), the variation of the action was
δS =
∫(δL)dτ +
∫L(δdτ)
=
∫ (∂L
∂ξαδξα +
∂L
∂uαδuα)dτ +
1
c2
∫Ldξαdτ
dδξα
=
∫ (∂L
∂ξαδξα +
∂L
∂uαdδξαdτ− 1
c2∂L
∂uαdξαdτ
dξβdτ
dδξβ
dτ
)dτ +
1
c2
∫Ldδξα
dτ
dξαdτ
dτ
=
∫ (∂L
∂ξα− d
dτ
(∂L
∂uα− 1
c2∂L
∂uβuβuα +
1
c2Luα
))δξαdτ
where uµ = dξµ/dτ . The equality between the third and fourth lines above is
given by integration by parts, again using the fact that the surface terms must
vanish due to the vanishing variations at the endpoints. Therefore, by demanding
δS = 0 and given that this must be satisfied for any arbitrary variations δξ, the
correct relativistic equations of motion are
d
dτ
(∂L
∂uα− 1
c2uα
(∂L
∂uβuβ − L
))=
∂L
∂ξα. (3.32)
70
3.5 Relativistic Constraints in Fluid Dynamics
Cavalleri and Spinelli first generalised Kalman’s result for a test particle in
general relativity [86] and then explored variations of an element of a perfect fluid
in special relativity [87]. While general relativistic theories are beyond the scope
of this thesis, the application of constrained variations to a special relativistic
perfect fluid will be reviewed here.
The total Lagrangian L is an integral of the Lagrangian density L over a
3-dimensional volume,
L =
∫LdV ≈ L∆V,
where ∆V is a fluid element (allowed to be compressible). Now the variation of
the action is
δS = δ
∫L∆V dτ =
∫(δL)∆V dτ +
∫L∆V (δdτ) +
∫L(δ∆V )dτ. (3.33)
The variations of L are in the same form as those for L, and the variation
of dτ was given in Eqn (3.30). Now it remains to find the δ∆V . Cavalleri &
Spinelli considered that the change in pressure P caused by varying ∆V must be
balanced by a change in the internal energy such that
Pδ(∆V ) = c2δ(ρ0∆V ) = c2(δρ0∆V + ρ0δ(∆V )) (3.34)
where ρ0 is the proper mass density. Rearranging terms in Eqn (3.34) gives
δ(∆V ) = − c2∆V
c2ρ0 + P
∂ρ0∂xα
δxα (3.35)
where the variation in mass density is
δρ0 =∂ρ0∂xα
δxα. (3.36)
A variation in the fluid volume element has now been taken into consideration
in varying the action and again demanding that δS = 0 gives the new equations
71
3.5 Relativistic Constraints in Fluid Dynamics
of motion
d
dτ
(∂L
∂uα+
1
c2uα
(L− ∂L
∂uβuβ
))=
∂L
∂xα−(∂L
∂uα+
1
c2uα
(L− ∂L
∂uβuβ
))u ,γγ
− c2L
c2ρ0 + p
∂ρ0∂xα
(3.37)
As expected, when the mass in the volume element ∆V remains constant, Eqn
3.37 reduces to Eqn 3.32. Cavalleri and Spinelli considered a Lagrangian density
of the form
L = c2ρ0gαβuαuβ + Pgαβu
αuβ
and substituting this into Eqn (3.37), using the continuity equation
(ρ0 +
P
c2
)uβ,β = −dρ0
dτ,
gives the equation of motion as
c
(ρ0 +
P
c2
)duαdτ
+1
c
dP
dτuα =
∂P
∂xα. (3.38)
However, while Cavalleri & Spinelli were discussing a fluid, the distinction be-
tween Eulerian and Lagrangian coordinates was not made clear in [87]. This was
compounded by the fact that Kalman’s original paper [85] and the generalisation
by Cavalleri & Spinelli themselves [86] both dealt with the Lagrangian coordinates
of a single test particle. It is unclear in Eqn (3.37) whether the four-velocity repre-
sents the Eulerian fluid velocity or the Lagrangian fluid particle velocity. Indeed,
the spacetime coordinates considered in the variation of the four-velocity and
proper time appear to be the same as those considered in the variation of the
mass density (3.36), which would imply an Eulerian four-velocity. However, as
was seen in §3.4, the variation of an Eulerian velocity is not so straight-forward,
so this author questions whether the variational principle resulting in (3.32) can
be applied without alteration to the case of a perfect fluid.
72
3.5 Relativistic Constraints in Fluid Dynamics
This author presents the following argument for a variation of the Eulerian
four-velocity in line with Newcomb’s method [79] for the three-dimensional non-
relativistic case. Consider the relativistic generalisation of Eqns (3.19) and (3.20)
such that the Eulerian four-velocity is related to the Lagrangian four-velocity
whereby
uα(xµ, τ)|xν=ξν(aω ,τ) =dξα(aµ, τ)
dτ(3.39)
and the virtual fluid displacement is
zα(xµ, τ)|xν=ξν(aω ,τ) = δξα(aµ, τ). (3.40)
The variation of Eqn (3.39) while following one fluid particle (that is, elimi-
nating the dependence of dξ/dτ on initial spacetime coordinate) is
δuα|xν=ξν(aω ,τ) = δ
(dξαdτ
)− δxβ ∂
∂xβ
(dξαdτ
),
This gives
δuα + zβ∂uα∂xβ
= δ
(dξαdτ
). (3.41)
Now taking the derivative of Eqn (3.40) with respect to proper time and
recalling Eqn (3.31,
dzαdτ
=dδξαdτ
= δ
(dξαdτ
)+
1
c2uαuβ
dδξβ
dτ. (3.42)
Substituting Eqns (3.40) and (3.42) into Eqn (3.41) gives the variation of the
Eulerian four-velocity for a fluid in terms of a virtual fluid displacement:
δuα =dzαdτ− 1
c2uαuβ
dzβ
dτ− zβ ∂uα
∂xβ. (3.43)
This differs from Kalman’s expression for the varied four-velocity by the last
term which takes into account the dependence of an Eulerian velocity on position
in space. Eqn (3.43) is however in agreement with another more general expres-
73
3.5 Relativistic Constraints in Fluid Dynamics
sion derived by Achterberg [90] which took into account curvilinear coordinates.
Now by substituting Eqns (3.30), (3.35) and (3.43) into Eqn (3.33), Hamilton’s
principle becomes, in terms of virtual fluid displacement
δS =
∫ (∂L
∂xαzα +
∂L
∂uα
(dzα
dτ− 1
c2uαuβ
dzβ
dτ− zβ ∂u
α
∂xβ
))∆V dτ
+
∫1
c2Luα
dzα
dτ∆V dτ −
∫c2L
c2ρ0 + P
∂ρ0∂xα
zα∆V dτ
Integrating by parts, and given that the surface terms vanish due to the virtual
fluid displacement zα disappearing at the endpoints, this becomes
δS =
∫ (∂L
∂xα− ∂L
∂uβ∂uβ
∂xα− 1
∆V
d
dτ
((∂L
∂uα− 1
c2uαu
β ∂L
∂uβ
)∆V
))zα∆V dτ
−∫ (
1
∆V
d
dτ
(1
c2Luα∆V
)− c2L
c2ρ0 + P
∂ρ0∂xα
)zα∆V dτ
Note that the additional term in Eqn (3.43) will now cancel with the explicit
derivative of L with respect to the Eulerian spacetime coordinates. Also, it is
true that
1
∆V
d∆V
dτ= uβ,β.
The requirement that δS = 0 and the arbitrariness of the virtual fluid dis-
placement finally gives the equation of motion
d
dτ
(∂L
∂uα− 1
c2uαu
β ∂L
∂uβ+
1
c2uαL
)= −
(∂L
∂uα− 1
c2uαu
β ∂L
∂uβ+
1
c2uαL
)uβ,β
− c2L
c2ρ0 + P
∂ρ0∂xα
,
which differs from Cavlleri & Spinelli’s original expression, (3.37), only in that the
term ∂L/∂xα is gone. In the equation of motion (3.38), this difference amounts
74
3.6 Maxwell-Vlasov System
to replacing the partial derivative of the pressure with a partial derivative of the
mass density such that
c
(ρ0 +
P
c2
)duαdτ
+1
c
dP
dτuα = −c2 ∂ρ0
∂xα. (3.44)
Relativistic laser-plasma interaction has also been investigated by Evstatiev
et al. [48] who used a Lagrangian formulation similar to that used by Brizard
[80] to formulate a relativistic Hamiltonian theory.
3.6 Maxwell-Vlasov System
A Maxwell-Vlasov system is described by three equations - Ampere’s Law and
Farday’s Law coupled with the Vlasov equation:
1
c2∂E
∂t= ∇×B− µ0qα
∫vfα(z, t)δ(z− z0)dz,
∂B
∂t= −∇× E,
∂fα∂t
= −v · ∂fα∂x− eαmα
∫(E + v ×B)
∂fα∂v
δ(z− z0)dz.
(3.45)
The index α identifies the charge, mass and distribution function for different
species of particles in the plasma. The last equation is the Vlasov equation, which
is simply Liouville’s theorem in disguised form. Given some particle distribution
function f(x,v, t), Liouville’s theorem states that the distribution function is
constant along its trajectory in phase space;
df
dt= v · ∂f
∂x+ a · ∂f
∂v+∂f
∂t= 0.
It is clear that the acceleration in the above equation can be replaced by the
Lorentz force expression divided by mass to give the Vlasov equation.
Morrison, Weinstein and Marsden showed that the Maxwell-Vlasov system of
75
3.6 Maxwell-Vlasov System
equations can be cast into the form of a continuous Hamiltonian system [55, 91,
92].
Given the Hamiltonian
H =
∫1
2mαv
2fαdz +
∫1
2(E2 +B2)dx
and a bilinear, anticommutative bracket that satisfies the Jacobi identity,
[F,G] =
∫f
{δF
δf,δG
δf
}dxdv +
∫ (δF
δE· ∇ × δG
δB− δG
δE· ∇ × δF
δB
)dx
+
∫δF
δE· ∂f∂v
δG
δE− δG
δE· ∂f∂V
δF
δfdxdv +
∫fB ·
(∂
∂v
δF
δf× ∂
∂v
δG
δf
)dxdv
where {F,G} is the canonical Poisson bracket given in Eqn (2.12), Eqns (3.45)
can be cast in Hamiltonian form so that
∂Bi
∂t= [Bi, H],
∂Ei∂t
= [Ei, H],∂fα∂t
= [fα, H].
3.6.1 Magnetohydrodynamics
A more general noncanonical Hamiltonian formalism was found for hydrodynam-
ics and magnetohydrodynamics (for an ideal fluid) by Morrison and Greene [57]
[56] in terms of the physical variables n, v, B and S (respectively fluid density,
velocity, magnetic field and entropy). The equations to be cast in Hamiltonian
form are
∂v
∂t= −∇v
2
2+ v × (∇× v)− 1
n∇(n2∂U
∂n
)+
1
n(∇×B)×B,
∂n
∂t= −∇ · (nv),
∂B
∂t= ∇× (v ×B),
∂S
∂t= −v · ∇S
(3.46)
76
3.6 Maxwell-Vlasov System
where U(n, S) is the internal energy per unit mass. This is similar to the set of
equations originally considered by Herivel [68] in §3.2, except that they are now
for the case of magnetohydrodynamics, that is, electrically conducting fluids.
Morrison and Greene directly took the Hamiltonian as corresponding to the
total energy density of the fluid,
H =1
2nv2 + nU +
1
2B2,
bypassing the need for a Lagrangian. The disadvantage to this approach is that
none of the variables are canonical in the sense that the usual form of Hamilton’s
equations will not give their time evolution. However, by defining a noncanonical
Poisson bracket (see §2.5),
[F,G] = −∫V
δF
δn∇ · δG
δv+δF
δv· ∇δG
δn+δF
δv·(∇× v
n× δG
δv
)(3.47)
+1
n∇S ·
(δF
δS
δG
δv− δG
δS
δF
δv
)+
1
n
δF
δv·(
B×(∇× δG
δB
))+δF
δB·(∇×
(B× 1
n
δG
δv
))dτ,
Eqns (3.46) take the familiar form
∂vi∂t
= [vi, H],∂Bi
∂t= [Bi, H],
∂n
∂t= [n,H],
∂S
∂t= [S,H]. (3.48)
All these equations can be transformed to Eulerian variables (usually preferred
for practical considerations, especially as in this case the fluid density appears in
the denominator of some terms in Eqn (3.47)) n, B, σ = nS and M = nv where σ
is the specific entropy and M is the momentum density. The transformed Poisson
77
3.7 Guiding Centre Motion
bracket is then
[F,G] = −∫V
n
(δF
δM· ∇δG
δn− δG
δM· ∇δF
δn
)+ M ·
(δF
δM· ∇ δG
δM− δG
δM· ∇ δF
δM
)+ σ
(δF
δM· ∇δG
δσ− δG
δM· ∇δF
δσ
)+ B ·
(δF
δM· ∇δG
δB− δG
δM· ∇δF
δB+
(∇δFδB
)· δGδM−(∇δGδB
)· δFδM
)dτ
The equations of motion are then in the same Hamiltonian form as in Eqns
(3.48), except that v is replaced by M and S by σ. Relativistic magnetohydro-
dynamics was explored by Achterberg [90] who applied Hamilton’s principle to a
relativistic Lagrangian to yield the same basic laws as those in (3.46), albeit in
relativistic form.
Several authors, most notably Morrison, have continued to apply Hamilto-
nian mechanics and Dirac’s constraint theory to hydrodynamics, magnetohydro-
dynamics and plasma physics in particular, with great success [47, 49, 93, 94].
3.7 Guiding Centre Motion
Several influential papers dealing with noncanonical Hamiltonian methods were
written by Littlejohn as well as Littlejohn & Cary [46, 59, 60]. Littlejohn first
considered a noncanonical Hamiltonian formulation of guiding centre motion [46],
which describes the motion of a charged particle in a magnetic field around a
central ‘guiding’ point which is itself moving. Such motion results in the par-
ticle following a helical orbit through space. This dynamic behaviour is most
certainly applicable to plasma physics, especially in MCF where the plasma is
confined in extremely strong magnetic fields, although it is not further considered
by this author beyond the following review (this thesis focuses on laser-plasma
interaction).
Littlejohn proposed that his Hamiltonian formulation was unique in preserving
Liouville’s theorem and allowing for time averaging of the guiding centre system
78
3.7 Guiding Centre Motion
(the relatively fast gyrating motion of the particle around the guiding centre is
most practically averaged out over one period). He considered a set of coordinates:
the particle position x; time t; kinetic energy k; u the component of velocity
parallel to the field; w the perpendicular velocity; θ the phase of the particle
gyration frequency about the guiding centre. All these coordinates are in the
reference frame moving with the guiding centre.
Note the introduction of time t as an independent variable, which is often
done when the canonical Hamiltonian explicitly depends on time. This‘extended
phase space’ has coordinates and conjugate momenta that now include time and
its conjugate variable, energy (k). The phase space trajectories are then all
parameterised by a new variable τ .
The Hamiltonian (derived by Littlejohn without need of the Legendre trans-
formation from a Lagrangian) is
H(x, t, k, u, w, θ) =1
2(u2 + w2)− k.
Littlejohn introduced the noncanonical Poisson bracket
[F,G] =∂F
∂qi{qi, qj}
∂G
∂qj
where qi represents all eight variables mentioned above and the bracket in the
centre is the usual Poisson bracket. The equations of motion then follow from the
noncanonical Poisson bracket of the generalised coordinates with the Hamiltonian
in the usual way where
∂qi∂τ
= [qi, H].
However, Littlejohn was not concerned with reproducing these equations as
much as applying Darboux’s theorem [54] (see §2.5) to find higher-order terms
in guiding centre motion, although the general concept of applying Hamiltonian
mechanics to guiding centre theory is of the greatest interest for the purposes of
this thesis.
79
3.8 The Korteweg-de Vries Equation
Littlejohn extended this idea to perturbation theory in noncanonical Hamil-
tonian coordinates [59], which was further expanded upon by Cary & Littlejohn
[60] in the context of magnetic field line flow.
Guiding centre theories were also investigated by Pfirsch & Morrison [95, 96]
who sought to find the explicit form of the energy-momentum tensor in this
context. However, in contrast to the material presented by this author in Chapter
4, Pfirsch & Morrison were not able to produce a manifestly gauge invariant
and symmetric tensor from their Lagrangian in [95]. It took some additional
arguments regarding the relation of the energy-momentum tensor to the angular
momentum tensor to prove symmetry. Gauge invariance was proven by splitting
the energy-momentum tensor into a gauge and non-gauge invariant part, and
then showing that the non-gauge invariant part was in fact zero. A linearised
Maxwell-Vlasov and kinetic guiding centre theories was addressed in [96] using a
Hamilton-Jacobi formalism and Dirac’s constraint theory (see §2.5). Brizard &
Tronko also looked at gyrokinetic conservation laws using Hamiltonian mechanics
[97].
An excellent review of these canonical formulations of guiding centre theories
can be found in [98].
3.8 The Korteweg-de Vries Equation
This section presents an interesting aside in the area of Hamiltonian fluid dy-
namics, specifically related to the Korteweg-de Vries (KdV) equation. The KdV
equation is a nonlinear wave equation;
∂u
∂t− u∂u
∂x− ∂3u
∂x3= 0.
Other versions of the KdV equation may include certain constant coefficients,
although these are always subject to renormalisation of the function u(x, t). This
equation has solutions that represent solitons or ‘solitary waves’. Solitons are a
80
3.8 The Korteweg-de Vries Equation
single wave packet propagating according to the above equation, with the inter-
esting property that they may pass through each other without interacting. They
are encountered in the study of plasma in the form of ion-acoustic waves.
A Hamiltonian formulation of the KdV equation was found by Gardner [99]
in the fourth of a series of papers [100, 101, 102, 103, 104] dedicated to the KdV
equation, its solutions, and its countably infinite conserved quantities.
Gardner found that given a Hamiltonian
H =
∫D
1
6u3 − 1
2
(∂u
∂x
)2
dx
and noncanonical Poisson bracket
{F,G} = −∫D
δF
δu
∂
∂x
δG
δudx,
the KdV equation emerges from Hamilton’s equation of motion for u;
∂u
∂t= {u,H} = −u∂u
∂x− ∂3u
∂x3.
The fact that the KdV equation has countably infinite integrals of motion is
due to the fact that its Hamiltonian formulation is not unique [105].
81
4
The Energy-Momentum Tensor
In Higher-Derivative Theories
4.1 The Gauge Invariant Electromagnetic Energy-
Momentum Tensor
This definition of the energy-momentum tensor creates difficulties in electro-
magnetism where the generalised coordinates are the components of the four-
potential. In this case, recalling Eqn (2.3), the tensor is defined as
T νµ =
∂L
∂Aα,νAα,µ − δνµL,
which is not a gauge invariant expression and therefore lacks the necessary phys-
ical interpretation of the components of the tensor. While this subject is treated
in every standard physics textbook , e.g, [42] [106], the gauge invariance of the
energy-momentum tensor of the free electromagnetic field is always treated after
calculating the tensor, in an ad hoc manner, by choosing to add a suitable diver-
genceless quantity. However, it was pointed out by Munoz [107], and much later
by Correa-Restrepo & Pfirsch [108], that this is not necessary and a careful treat-
ment of the problem can yield a manifestly gauge invariant expression, provided
82
4.1 The Gauge Invariant Electromagnetic Energy-Momentum Tensor
that the Lagrangian itself is gauge invariant.
The following derivation of the free electromagnetic field energy-momentum
tensor follows Munoz, but it will be extended by this author in §4.2 to include
Lagrangians with derivatives of higher-order. Consider the action
S =
∫L(Aµ, Aµ,ν)d
4x. (4.1)
The Principle of Least Action gives the usual Euler-Lagrange equations
∂L
∂Aµ=
∂
∂xα
(∂L
∂Aµ,α
).
Consider now an infinitesimal transformation of the coordinates such that
x′τ = xτ + ωτσxσ + aτ = xτ + ετ (x)
where ωτσ represents a Lorentz transformation (and so is necessarily an anti-
symmetric tensor) and aτ is an infinitesimal local translation of the coordinates.
With this transformation,
A′µ =∂xα
∂x′µAα = (δαµ −
∂εα
∂x′µ)Aα = (δαµ −
∂xβ
∂x′µ
∂εα
∂xβ)Aα. (4.2)
This expression can be evaluated to any order in
∂xβ
∂x′µ,
given the appearance of this expression again on the right hand side. Taken to
first order, we have
A′µ = (δαµ − εα,µ)Aα. (4.3)
Similarly, for first-order derivatives of the potential,
A′µ,ν = (∂ν − εβ,ν∂β)(δαµ − εα,µ)Aα (4.4)
83
4.1 The Gauge Invariant Electromagnetic Energy-Momentum Tensor
A change of coordinates in Eqn (4.1) gives
S ′ =
∫L(A′µ(x′),
∂
∂x′νA′µ(x′))d4x′
=
∫L(Aµ − εα,µAα, (∂ν − εβ,ν∂β)(δαµ − εα,µ)Aα
)d4x
If the variation of S is now calculated using the fact that the variation in the
coordinates Aµ is given by the infinitesimal transformation part of Eqn (4.3) to
first order in ε, then
δS = S ′ − S = −∫ (
∂L
∂Aµεα,µAα +
∂L
∂Aµ,ν(εα,µAα,ν + εα,νAµ,α)
)d4x. (4.5)
If the Lagrangian itself is gauge invariant, as in the case of the free electro-
magnetic field (but not where there is an interaction term for a particle in a field),
then the derivatives of the Lagrangian with respect to the field must be gauge
invariant and appear only in the combination
Aµ,ν − Aν,µ.
In this case, it is true that
∂L
∂Aµ,ν= − ∂L
∂Aν,µ.
Eqn (4.5) then becomes
δS = −∫ (
∂L
∂Aµεα,µAα −
∂L
∂Aν,µεα,µAα,ν +
∂L
∂Aµ,νεα,νAµ,α
)d4x.
84
4.1 The Gauge Invariant Electromagnetic Energy-Momentum Tensor
Now a simple relabelling of the dummy indices µ, ν in the second term gives
δS = −∫ (
∂L
∂AνAα +
∂L
∂Aµ,ν(Aµ,α − Aα,µ)
)εα,νd
4x
= −∫∂ν
((∂L
∂AνAα +
∂L
∂Aµ,ν(Aµ,α − Aα,µ)
)εα)d4x
+
∫∂ν
(∂L
∂AνAα +
∂L
∂Aµ,ν(Aµ,α − Aα,µ)
)εαd4x
Call the expression inside the brackets in the second term I να and evaluate
its divergence using: the Euler-Lagrange equations; independence of L from Aµ;
antisymmetry of the derivatives with respect to Aµ,ν . This yields
I να,ν =
∂
∂xν
(∂L
∂Aν
)Aα +
∂L
∂AνAα,ν +
∂
∂xν
(∂L
∂Aµ,ν
)(Aµ,α − Aα,µ)
+∂L
∂Aµ,ν(Aµ,αν − Aα,µν)
=∂
∂xν
(∂L
∂Aµ,νAµ,α
)− ∂L
∂Aµ,νAα,µν
=∂
∂xν
(∂L
∂Aµ,νAµ,α
)
If the derivation that led to Eqn (2.3) were repeated with respect to a partial
derivative of L, then the result would be
∂L
∂xα=
∂
∂xν
(∂L
∂Aµ,νAµ,α
)= I ν
α,ν .
The variation of the action is then
δS = −∫ (
∂ν
((∂L
∂AνAα +
∂L
∂Aµ,ν(Aµ,α − Aα,µ)
)εα − ενL
))d4x
since εν,ν = 0 due to the antisymmetry of the Lorentz transformation ωντ . The ap-
plication of Noether’s theorem is now a simple matter, and the conserved current
85
4.2 Lagrangians with Higher-Order Derivatives
is
Jν =
(∂L
∂AνAα +
∂L
∂Aµ,ν(Aµ,α − Aα,µ)
)εα − ενL
where the energy-momentum tensor belongs to the translational part of ε, and the
angular momentum tensor to the rotational part (the Lorentz transformation).
A manifestly gauge invariant expression for the energy-momentum tensor is then
simply expressed as
T να = 2∂L
∂FνµFαµ − δναL (4.6)
since
∂L
∂Aν= 0
for a gauge invariant Lagrangian and derivatives with respect to the four-potential
can be replaced with derivatives with respect to the electromagnetic tensor;
∂L
∂Aα,β=
∂Fτω∂Aα,β
∂L
∂Fτω(4.7)
= (δβτ δαω − δατ δβω)
∂L
∂Fτω
= 2∂L
∂Fβα.
4.2 Lagrangians with Higher-Order Derivatives
The derivation of the canonical energy-momentum tensor (2.3) can be generalised
for higher derivatives, as can the Euler-Lagrange equations themselves. More
generally, the Euler-Lagrange equations can be found as a functional derivative
of the action with respect to the generalised coordinates qi. Such a functional
derivative includes all higher derivatives of the generalised coordinates that may
appear in the Lagrangian (and therefore the action), and is given by
δS
δqi=
∫ (∂L
∂qi− d
dxα
(∂L
∂qi,α
)+
d2
dxαdxβ
(∂L
∂qi,αβ
)− . . .
)d4x.
If the variation of S is zero for any arbitrary δqi, then the above expression
86
4.2 Lagrangians with Higher-Order Derivatives
implies the higher-derivative Euler-Lagrange equations
=⇒ ∂L
∂qi=
d
dxα
(∂L
∂qi,α
)− d2
dxαdxβ
(∂L
∂qi,αβ
)+ . . . (4.8)
Therefore, if the Lagrangian includes second-order derivatives of the gener-
alised coordinates, the derivation of the energy-momentum tensor follows the
same procedure as in §4.1, but this time substituting (4.8);
dL
dxµ=∂L
∂qiqi,µ +
∂L
∂qi,νqi,νµ +
∂L
∂qi,βνqi,βνµ +
∂L
∂xµ
=d
dxν
(∂L
∂qi,ν
)qi,µ −
d2
dxνdxβ
(∂L
∂qi,νβ
)qi,µ +
∂L
∂qi,νqi,νµ +
∂L
∂qi,βνqi,βνµ +
∂L
∂xµ
=d
dxν
(∂L
∂qi,νqi,µ
)− d
dxν
(d
dxβ
(∂L
∂qi,νβ
)qi,µ
)+
d
dxβ
(∂L
∂qi,νβ
)qi,µν
+d
dxν
(∂L
∂qi,βνqi,βµ
)− d
dxν
(∂L
∂qi,βν
)qi,βµ +
∂L
∂xµ
=d
dxν
(∂L
∂qi,νqi,µ −
d
dxβ
(∂L
∂qi,νβ
)qi,µ +
∂L
∂qi,βνqi,βµ
)+∂L
∂xµ
=⇒− ∂L
∂xµ=
d
dxν
(∂L
∂qi,νqi,µ −
d
dxβ
(∂L
∂qi,βν
)qi,µ +
∂L
∂qi,βνqi,βµ − δνµL
).
The energy-momentum tensor is then
T νµ =∂L
∂qi,νqi,µ −
d
dxβ
(∂L
∂qi,βν
)qi,µ +
∂L
∂qi,βνqi,βµ − δνµL. (4.9)
However, if the generalised coordinates qi represent the components of a
vector-field, as in the case of the electromagnetic potential, then the above ex-
pression is not manifestly gauge invariant. To remedy this, the same procedure
demonstrated by Munoz which was reviewed in §4.1 will now be applied, for the
first time, to a gauge invariant Lagrangian dependent on second-order derivatives
of the electromagnetic potential. Consider the action integral
S =
∫L(Aµ, Aµ,ν , Aµ,ντ )d
4x
87
4.2 Lagrangians with Higher-Order Derivatives
subject to an infinitesimal transformation of coordinates such that
x′τ = xτ + ωτσxσ + aτ = xτ + ετ (x),
then
∂2A′(x′)µ∂x′τ∂x′ν
=∂xα
∂x′µ
∂xβ
∂x′ν
∂xγ
∂x′τAα,βγ
= (δαµ −∂εα
∂x′µ)(δβν −
∂εβ
∂x′ν)(δγτ −
∂εγ
∂x′τ)Aα,βγ.
According to Eqns (4.2) and (4.3), this gives, to first order in ε,
∂2A′(x′)µ∂x′τ∂x′ν
= (δαµ − εα,µ)(δβν − εβ,ν)(δγτ − εγ,τ )Aα,βγ
= (δαµδβν δ
γτ − δαµδβν εγ,τ − δβν δγτ εα,µ − δαµδγτ εβ,ν)Aα,βγ
= Aµ,ντ − εγ,τAµ,νγ − εα,µAα,ντ − εβ,νAµ,βτ .
The variation of the action given by S ′ − S is then
δS = −∫ (
∂L
∂Aµεα,µAα +
∂L
∂Aµ,ν(εα,µAα,ν + εα,νAµ,α)
)d4x
−∫ (
∂L
∂Aµ,ντ(εγ,τAµ,νγ + εα,µAα,ντ + εβ,νAµ,βτ )
)d4x.
Relabelling some dummy indices gives
δS = −∫ (
∂L
∂AµAα +
∂L
∂Aµ,νAα,ν +
∂L
∂Aν,µAν,α
)εα,µd
4x (4.10)
−∫ (
∂L
∂Aτ,νµAτ,να +
∂L
∂Aµ,ντAα,ντ +
∂L
∂Aν,µτAν,ατ
)εα,µd
4x.
88
4.2 Lagrangians with Higher-Order Derivatives
Since the Lagrangian is gauge invariant, derivatives of L with respect to
derivatives of the four-potential can be replaced by derivatives with respect to
the electromagnetic tensor as follows:
∂L
∂Aα,βγ=
∂F τω,ω
∂Aα,βγ
∂L
∂F τε,ε
(4.11)
=∂(Aω,τω − Aτ,ωω)
∂Aα,βγ
∂L
∂F τε,ε
=∂L
∂F ,εβε
gαγ − ∂L
∂F ,εαεgγβ.
While it is clear from the above expression that
∂L
∂Aα,βγ= − ∂L
∂Aβ,αγ,
extreme care must still be taken with the order of the four-potential derivatives
appearing in such expressions, since they also satisfy a kind of Jacobi identity
whereby
∂L
∂Aα,βγ+
∂L
∂Aγ,αβ+
∂L
∂Aβ,γα= 0.
Consider the terms inside the brackets in (4.10) and call them I µα collectively,
so that
δS =
∫ (I µα,µε
α − ∂
∂xµ(I µα ε
α)
)d4x. (4.12)
This can be cast into a form allowing the application of Noether’s theorem as
follows. First, consider the partial derivative of the Lagrangian,
∂L
∂xα=
∂L
∂AνAν,α +
∂L
∂Aν,µAν,µα +
∂L
∂Aν,µτAν,µτα.
Substituting the Euler-Lagrange equations in the first term, then simplifying,
gives
∂L
∂xα=
∂
∂xµ
(∂L
∂Aν,µAν,α −
∂
∂xτ
(∂L
∂Aν,µτ
)Aν,α +
∂L
∂Aν,τµAν,τα
).
89
4.2 Lagrangians with Higher-Order Derivatives
Comparing the above expression with I µα,µ, we have
∂L
∂xα= I µ
α,µ −∂
∂xµ
(∂L
∂AµAα +
∂L
∂Aµ,νAα,ν +
∂L
∂Aµ,ντAα,ντ +
∂L
∂Aν,µτAν,ατ
)− ∂
∂xµ
(∂
∂xτ
(∂L
∂Aν,µτ
)Aν,α
).
Using the Euler-Lagrange equations again, the antisymmetry of the derivatives
of L with respect to the four-potential derivatives, and the Jacobi identity for
derivatives with respect to the second order derivatives of the four-potential, the
above expression simplifies to
I µα,µ =
∂L
∂xα+
∂2
∂xµ∂xτ
(∂L
∂Aµ,ντFνα
).
The second order derivative of ε vanishes and εα,α = 0, so therefore
I µα,µε
α =∂Lεα
∂xα+
∂
∂xµ
(∂
∂xτ
(∂L
∂Aµ,ντFνα
)εα − ∂L
∂Aτ,νµFναε
α,τ
)
The third term in the expression above can be expressed with a factor of ε, and
not its derivative, by use of Eqn (4.4), which gives:
I µα,µε
α =∂Lεα
∂xα+
∂
∂xµ
(∂
∂xτ
(∂L
∂Aµ,ντFνα
)εα +
∂L
∂Aτ,νµ(A′ν,τ − Aν,τ )
)(4.13)
=∂Lεα
∂xα+
∂
∂xµ
(∂
∂xτ
(∂L
∂Aµ,ντFνα
)εα +
1
2
∂L
∂Aτ,νµFντ,αε
α
).
Now let
I µα =
∂L
∂AµAα + 2
∂L
∂FνµFνα +K µ
α (4.14)
90
4.2 Lagrangians with Higher-Order Derivatives
where
K µα =
∂L
∂Aτ,νµAτ,να +
∂L
∂Aµ,ντAα,ντ +
∂L
∂Aν,µτAν,ατ
=1
2
∂L
∂Aτ,νµFντ,α +
∂L
∂Aµ,ντFνα,τ
Substituting (4.13) and (4.14) into (4.12), and then applying (4.11), finally gives
the variation of the action as
δS = −∫
∂
∂xµ
((∂L
∂AµAα + 2
∂L
∂FνµFνα
)εα)d4x
−∫
∂
∂xµ
((− ∂
∂xµ
(∂L
∂F ,βνβ
)Fνα +
∂
∂xν
(∂L
∂F ,βµβ
)Fνα − δµαL
)εα
)d4x
The conserved current is therefore
Jµ =
(∂L
∂AµAα + 2
∂L
∂FνµFνα −
∂
∂xµ
(∂L
∂F ,σνσ
)Fνα +
∂
∂xν
(∂L
∂F ,σµσ
)Fνα − δµαL
)εα
with the translational part of ε giving the energy-momentum tensor. So the final
gauge invariant energy-momentum tensor for a gauge invariant electromagnetic
Lagrangian dependent on second-order derivatives is
T νµ = 2∂L
∂FναFµα −
∂
∂xν
(∂L
∂F ,σασ
)Fαµ +
∂
∂xα
(∂L
∂F ,σνσ
)Fαµ − δνµL. (4.15)
However, if the Lagrangian itself is not gauge invariant (as in the case where
there is an interaction term for a charged particle, JαAα), then (4.15) can still
be used to simplify calculations by splitting it into a gauge invariant part and
non-gauge invariant part. The non-gauge invariant part must then be dealt with
using the canonical expression (4.9) such that
T νµ =∂L
∂Aα,νAα,µ −
d
dxβ
(∂L
∂Aα,βνAα,µ
)+ 2
∂L
∂Aα,βνAα,βµ − δνµL. (4.16)
This thesis has therefore demonstrated, for the first time, that the rigorous
91
4.2 Lagrangians with Higher-Order Derivatives
approach of Munoz is applicable to a second order theory and yields an expression
for the energy-momentum tensor that is manifestly gauge invariant.
It is well worth mentioning that Lagrangians with higher derivatives were orig-
inally studied by Ostrogradski [109] who considered the stability of the Hamil-
tonian corresponding to such a Lagrangian. While Ostrogradski’s work has seen
some application in the world of physics [110, 111, 112], it will not be further
explored in this thesis beyond the following precis of Ostrogradski’s work.
Ostrogradski showed that if the highest time derivative of the Lagrangian is
non-degenerate, the Hamiltonian will have at least one linear instability [112].
The term non-degenerate means that the equations that give the canonical mo-
mentum Pi,
Pi =δL
δqi,
can be inverted to give qi in terms of the canonical coordinates only. Otherwise,
the Legendre transform of the Lagrangian (the Hamiltonian),
H = Piqi − L(qi, qi),
could not be expressed purely in terms of canonical coordinates Qi, Pi. The linear
instability refers to the appearance of a term linear in the canonical momentum
in the Hamiltonian, rather than quadratic as usual, therefore leaving the Hamil-
tonian (energy) unbounded from below.
While Ostogradski’s theorem is still be a focus of study for any physicist ex-
ploring Hamiltonian descriptions of nature, the following section will deal purely
with the Lagrangian side of the higher-derivatives problem. In particular, the
Podolsky Lagrangian will be substituted into (4.15) to give a gauge invariant
expression for the Podolsky energy-momentum tensor.
92
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
4.3 The Gauge Invariant Energy-Momentum Ten-
sor for the Podolsky Lagrangian
Possibly the most well-known example of a Lagrangian theory of fields with higher
derivatives is that of Podolsky electrodynamics. Podolsky considered Ostragrad-
ski’s approach applied to electromagnetism and found that there was only one
possible generalisation of the electromagnetic Lagrangian that would yield linear
field equations below sixth-order [7];
L = − 1
4µ0
FαβFαβ +
1
2µ0
a2Fαβ,βF
,γαγ (4.17)
where a is some new constant of nature with dimensions of length. This constant
will be addressed in more detail toward the end of this section, but for now it is
sufficient to say that this constant must be very small, otherwise the deviations
from Maxwell’s theory predicted by Podolsky would have been detected already.
Cuzinatto et al. [113] showed that, to be consistent with quantum theory and
experiments, a ≤ 5.6 × 10−15m - the order of the Compton wavelength of the
electron [114].
That (4.17) is the only possible modification of the electromagnetic Lagrangian
(giving linear differential equations below sixth-order) can be seen by considering
the underlying nature of the electromagnetic tensor in terms of the electric and
magnetic fields. Podolsky found that the extra term in (4.17) is the only other
Lorentz invariant that can be constructed with the electromagnetic tensor that
does not make a vanishing contribution to the field equations (while E ·B is also a
Lorentz invariant, it can be expressed as the four-divergence of the four-current,
and would therefore not contribute anything to the equations of motion). The
uniqueness of (4.17) has also been re-examined and confirmed recently [115].
The Podolsky Lagrangian keeps Maxwell’s electrodynamics largely intact in
that the Euler-Lagrange equations are simply generalisations of Maxwell’s equa-
tions that take into account higher-derivatives. The Lorentz force is also un-
93
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
changed by this Lagrangian, since it is derived from the kinetic energy and in-
teraction term in the Lagrangian, which can be added as usual to (4.17). What
then is the point of the Podolsky Lagrangian? The critical difference between
Podolsky’s Lagrangian and the standard electromagnetic Lagrangian is that it
gives a finite energy for a charged point particle without any further tinkering
or renormalisation. It also better describes waves of very high frequency where
derivatives of the field can no longer be considered negligible, and can be effec-
tively quantized [116].
The Euler-Lagrange equations for the Lagrangian (4.17) are
δL
δAµ= 0 =⇒ ∂L
∂Aµ=
∂
∂xτ
(∂L
∂Aµ,τ
)− ∂2
∂xωxτ
(∂L
∂Aµ,τω
)
which gives (1 + a2
∂2
∂xγxγ
)F ,τµτ = 0 (4.18)
Converting F ,τµτ to vector notation, Eqn (4.18) is more easily recognisable as
a generalised version of Maxwell equations in free space:
(1 + a2
∂2
∂xγ∂xγ
)(∇×B− 1
c2∂E
∂t
)= 0; (4.19)
(1 + a2
∂2
∂xγ∂xγ
)∇ · E = 0. (4.20)
Including the current and charge densities on the right hand side of Eqn (4.19)
and (4.20), respectively, only requires the addition of an interaction term JαAα
to the Lagrangian, where Jα is the four-current. In electrostatics,
E = −∇φ,
and since the electric field does not change, the second order time derivative in
Eqn (4.20) can be ignored. The generalised Poisson’s equation for a point charge
94
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
qδ(r), where δ is the Dirac-delta functional, is then
(1 + a2∇2)∇2φ = − 1
ε0qδ(r).
This equation has only one solution of the form
φ =q(1− e−r/a)
ε0r
where r is now the scalar radial distance from the origin. Note that by application
of l’Hopital’s Rule,
limr→0
q(1− e−r/a)ε0r
= limr→0
q
ε0ae−r/a =
q
ε0a,
indicating that the self-energy of a charged point particle has a finite value at
q2/ε0a, unlike standard electrodynamics where the energy diverges as r → 0.
The Lagrangian (4.17) will now be subsituted into Eqn (4.15 which gives the
energy-momentum tensor for Podolsky electrodynamics:
T νµ = Tνµ +a2
µ0
(−Fασ ν
,σ F µα + F νσ α
,σ F µα − gνµ
1
2Fαβ
,βF,γ
αγ
)(4.21)
where Tνµ is the usual Maxwell energy-momentum tensor for the unmodified
Lagrangian [106],
Tνµ =1
µ0
(−F ναF µα + gνµ
1
4FαβF
αβ).
The gauge invariant energy-momentum tensor (4.15) was derived in general for
any gauge invariant Lagrangian and when applied to the Podolsky Lagrangian
it yielded (4.21) - a tensor that is already gauge invariant, in contrast to that
derived by Podolsky himself [7] [117] [118].
To satisfy the reader that (4.21) is indeed correct, its equivalence to the origi-
nal Podolsky tensor will be demonstrated. Note that the second and third terms
can be combined via the Jacobi identity for the electromagnetic tensor such that
95
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
−Fασ ν,σ F µ
α + F νσ α,σ F µ
α = F να,σσF
µα . (4.22)
Now consider an odd expression that appears to come from nowhere, but will
provide a useful (but not obvious) identity when the Podolsky Lagrangian (4.17)
is inserted;
∂L
∂Fντ,αFατ,µ =
1
2
∂L
∂Fντ,αFατ,µ −
1
2
∂L
∂Fτν,αFατ,µ (4.23)
F νσ,σF
ττ,µ =
1
2F νσ
,σFττ,µ +
1
2F τσ
,σFντ,µ
=⇒ ∂
∂xµ(F ,σ
ασ Fαν)− F ,σµ
ασ Fαν = F ,σασ F
αν,µ = 0 (since Fαα = 0).
The divergence of the first term of the last line above is
∂2
∂xν∂xµ(F ,σ
ασ Fαν) =
∂2
∂xβ∂xµ(F ,σ
ασ Fαβ) =
∂2
∂xν∂xβ(gµνF ,σ
ασ Fαβ)
=⇒ ∂
∂xµ(F ,σ
ασ Fαν) =
∂
∂xβ(gµνF ,σ
ασ Fαβ).
Also, note that
gµνF σαβ,σ F
αβ = −gµνF σσα,β F
αβ − gµνF σβσ,α F
αβ
= 2∂
∂xβ(gµνF ,σ
ασ Fαβ)− 2gµνF ,σ
ασ Fαβ,β
which means that
∂
∂xµ(F ,σ
ασ Fαν) =
∂
∂xβ(gµνF ,σ
ασ Fαβ) = gµν(
1
2F σαβ,σ F
αβ + F ,σασ F
αβ,β). (4.24)
Substituting (4.24) for the first term of the last line in (4.23), and using the
96
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
Jacobi identity for the electromagnetic tensor on the second term, gives
F ,σασ F
αν,µ = gµν(1
2F σαβ,σ F
αβ + F ,σασ F
αβ,β) + F µα,σ
σFνα + F σµ,α
σFνα
= gµν(1
2F σαβ,σ F
αβ + F ,σασ F
αβ,β)− F µα,σ
σFνα +
∂
∂xα(F µσ
,σFνα)
− F µσ,σF
,ανα (4.25)
The fourth term in the second line above is divergenceless, so substituting
(4.22) and (4.25) (which is equal to zero and does not affect the equations of
motion) back into (4.21) gives the symmetric energy-momentum tensor as it was
originally derived by Podolsky:
T νµ = Tνµ +a2
µ0
(gµν1
2(F σ
αβ,σ Fαβ + F ,σ
ασ Fαβ,β)− F να,σ
σFµα − F µα,σ
σFνα
− F νσ,σF
µα,α). (4.26)
There are much simpler ways to make Eqn (4.21) symmetric, but the goal here was
to prove an exact match to the symmetric version originally derived by Podolsky
who used a different method to construct the tensor.
Since the focus of this thesis is on plasma physics, an investigation of the
parameter a and the dispersion relation for waves in a plasma modeled on the
Podolsky Lagrangian will be considered. Would the Debye length then be an
appropriate value to assume for the constant a, or is there another way to derive
a for laser-plasma interaction without further assumption?
Consider that if a current is present then Eqn (4.19) is
(1 + a2
∂2
∂xγxγ
)(∇×B− 1
c2∂E
∂t
)= µ0J.
Even though the Podolsky Lagrangian contains higher derivatives of the field,
it assumed no modification to the particle and particle/field interaction terms in
the Lagrangian. In this case, the Lorentz force is unchanged and the generalised
97
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
Ohm’s law for a plasma, Eqn (1.10) derived in §1.2.5 is still applicable where
J = iε0ω2p
ωE.
A new plasma dispersion relation then results from
(1 + a2
∂2
∂xγxγ
)(∇×B− 1
c2∂E
∂t
)= i
ω2p
c2ωE
where E and B are assumed proportional to ei(k·r−ωt) as in §1.2.5. The dispersion
relation is (k2 − ω2
c2
)(1 + a2
(k2 − ω2
c2
))= −
ω2p
c2, (4.27)
which can be rearranged to give a in terms of the variables ωp, k and ω
a2 =−ω2
p − c2k2 + ω2
c2(k2 − ω2/c2)2.
Alternatively, it could also be expressed in the same form as Eqn (1.4) with
(squared) dimensions of speed over frequency;
a2 =c2 + ω2
p/k2 − ω2/k2
−c2/k2(k2 − ω2/c2)2. (4.28)
Eqn (4.28) has been presented for the first time in this thesis, and no-one
else has attempted deriving the Podolsky constant in this way in the context of
laser-plasma interaction.
An analysis of the above expression can yield values for the frequencies that
could probe Podolsky electrodynamics. It is clear that the numerator can be
at most of order 9 × 1016m2s−2 since ω > ωp or the wave would be cut off. It
was already mentioned at the beginning of this section that a ≤ 5.6 × 10−15m.
Therefore, to be consistent with the femtometre scale of a, the denominator must
be ≥ 2.9 × 1045s−2, which corresponds to an extremely high energy wave in the
gamma ray range with a frequency of ≈ 5.4× 1022s−1 or higher. Therefore, high
98
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
intensity laser-plasma interaction may provide a testbed for the predictions of
Podolsky electrodynamics.
Note that in Maxwell’s electrodynamics, ω2 = c2k2, which is meaningless
in Podolsky electrodynamics since a would be undefined. That the dispersion
relation for electromagnetic waves is altered in Podolsky electrodynamics, such
that ω2 6= c2k2, implies that photons have mass. Of course, this mass must be
extremely small to fit experimental data which has set an upper mass on the
photon at between 7× 10−17eV/c2 [119] and 3× 10−27eV/c2 [120], depending on
the model used and assumptions made about the photon itself.
Contrary to popular belief, a massive photon does not preclude gauge invari-
ance in electrodynamics [121]. Podolsky Electrodynamics is a gauge and Lorentz
invariant theory that incorportates photon mass. However, most authors overlook
this since the Proca Lagrangian, given by
L = − 1
4µ0
FαβFαβ + µAαA
α,
is more well-known than the Podolsky Lagrangian and is usually the model used
to incorporate photon mass µ into electrodynamics (at the expense of gauge
invariance).
A link has been previously made between Podolsky electrodynamics and
plasma physics by Santos [122]. He noted that the modified Podolsky disper-
sion relation (for free space),
(k2 − ω2
c2
)(1 + a2
(k2 − ω2
c2
))= 0, (4.29)
indicates that the electromagnetic waves are travelling through a tenuous plasma
even though the free space solution is being considered. To see this, recall Eqn
(1.4) for the Debye length derived in §1.2.2,
λD =vavgωp
.
99
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
Given that the dispersion relation in (4.29) was derived for vacuum, the av-
erage velocity in the equation above can be identified with c while the constant
a (with dimensions of length) can be identified with the Debye length, giving an
expression for the frequency of a plasma in ‘free space’,
a =c
ωp.
In this way, Santos assumed a form for the constant a, as opposed to the
method presented in this thesis that led to Eqn (4.27) by deriving it from first
principles. In this case, Eqn (4.29) gives two modes for wave propagation in free
space; the standard free space dispersion relation
ω = ck
and the plasma dispersion relation
ω2p = ω2 − c2k2.
Santos believes this plasma relation to be due to vacuum polarization which
creates a particle density given by
n =ε0mω
2p
q2=
m
µ0q2a2≈ 760× 1036m−3
where a = λC/2. Given that the reduced Compton wavelength λC of an electron
is approximately 3.86× 10−13m [114], a sphere of radius λC has a total volume of
roughly 2.409× 10−37m3 which would contain on average around 183 virtual par-
ticles according to Podolsky electrodynamics. Santos believes that the creation
and annihilation of these particles contributes to the Zitterbewegung, or electron
‘jitter’ motion predicted by quantum mechanics at distances of the order of the
Compton wavelength [123].
Whether either Santos or this author (or both) are correct in their analysis of
100
4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian
Podolsky electrodynamics is a question that can only be settled by experiment.
However, the point to emphasise here is that the length and energy scales dis-
cussed in this section are in the realm of high intensity laser-plasma interactions
encountered in ICF experiments. This thesis posits that the seemingly small
modification of the electromagnetic Lagrangian introduced by Podolsky will in
fact yield more accurate predictions in laser-plasma interaction.
101
5
The Nonlinear Ponderomotive
Force in Laser-Plasma Interaction
5.1 The Ponderomotive Force
A ponderomotive force is usually defined as a force on a charged particle due to
an oscillating electric field, expressed as
f = − q2
4mω2∇E2
where ω is the oscillation frequency of the field and m the mass of the particle.
This force was first observed in experiment in 1957 [124].
The derivation of the ponderomotive force is reasonably straightforward and
uses the Lorentz force with the assumption of a varying electric field amplitude.
Consider first the Lorentz force for an electromagnetic field oscillating with fre-
quency ω such that
mdv
dt= q(E + v ×B) cosωt.
Neglecting the magnetic field for the moment gives a first approximation of
the particle velocity as
v(1) =qE sinωt
mω(5.1)
102
5.1 The Ponderomotive Force
and position as
r(1) = −qE cosωt
mω2. (5.2)
These approximations will be used as first order variations in such a way that
a small variation in the electric field amplitude is given by
δE = δxi∂E
∂xi= (r(1) · ∇)E,
which also has the effect of altering the velocity where
δv = v(1).
With this variation in the electric field amplitude, the Lorentz force is given
by
f = q((r(1) · ∇)E + v(1) ×B) cosωt (5.3)
= −q2 cos2 ωt
mω2(E · ∇)E +
q2 sinωt cosωt
mωE×B
Although the field is oscillating, the cumulative effect of the force on a charged
particle over time is of real interest. Consider the average of cos2 ωt over time;
〈cos2 ωt〉 = limT→∞
1
T
∫ T
0
(1
2cos 2ωt+
1
2)dt
= limT→∞
1
T
[1
4ωsin 2ωt+
t
2
]T0
=1
2.
The same procedure gives
〈sinωt cosωt〉 = 〈12
sin 2ωt〉 = 0,
103
5.1 The Ponderomotive Force
and so the time averaging of Eqn (5.3) gives
f = − q2
2mω2(E · ∇)E.
Using the vector calculus identity
E · ∇E =1
2∇E2 − E× (∇× E)
and the Maxwell-Faraday equation
∇× E = −∂B
∂t
gives
f = − q2
2mω2
(1
2∇E2 + E× ∂B
∂t
).
However, considering again the time average of this expression means that
B does not change in time, so the second term above is zero, finally giving the
Ponderomotive force as
f = − q2
4mω2∇E2. (5.4)
Interestingly, the direction of this force does not depend on the sign of the
charged particle itself. This means that in a plasma, both electrons and ions
would be accelerated in the same direction by the ponderomotive force, although
the much greater mass of the ions limits their acceleration relative to the plasma
electrons. Note that the ponderomotive force causes charged particles to drift
toward areas of low field intensity, pushing them outward from a laser beam.
This is in contrast to similar force derivations applied to dielectric molecules in a
laser beam which are accelerated toward regions of higher intensity according to
f =1
2α∇E2,
104
5.2 The Nontransient Nonlinear Ponderomotive Force
where the polarisation P of the molecule satisfies the linear relation
P = αE
and α has the same dimensions as ε0. In this case, the molecules are trapped in
the laser beam which acts as an ‘optical tweezer’ [125].
The significance of the ponderomotive force in relation to high intensity laser
beams was explored by Quesnel and Mora [126]. They showed that at high
intensities, the force must be generalised to a relativistic ponderomotive force. A
generalisation of the transient nonlinear ponderomotive force will be discussed in
§5.3.
5.2 The Nontransient Nonlinear Ponderomotive
Force
The nonlinear ponderomotive force is particularly relevant in the context of iner-
tial confinement fusion. Indeed, the very promising side-on ignition fusion scheme
has a theoretical foundation in the nonlinear force [127, 128]. The prospect of
fusion energy production with less radioactivity than burning coal (even coal
contains trace amounts of uranium) via Hydrogen/Boron-11 side-on ignition is
exciting given recent calculations suggesting that this reaction would be only one
order of magnitude more difficult than traditional Deuterium/Tritium reactions
using spherical laser compression [129, 130].
It was shown by Schluter [131] that a simple two-fluid model of a plasma
could yield an equation of motion for the plasma current in terms of the pon-
deromotive force. An in-depth discussion of this can be found in [18], bu the
essential argument uses the two-fluid model of a plasma. Assuming only one ion
species in the plasma, ions and electrons can be individually treated as charged
fluids. The equations of motion for each plasma particle species is then given by
105
5.2 The Nontransient Nonlinear Ponderomotive Force
a combination of the Euler equation with the Lorentz force. In the case of ions,
the equation of motion in vector notation is
minidvidt
= −ZniqE− niZqvi ×B−∇(nikBTi)−mneνei(vi − ve) (5.5)
where the subscript i denotes that these quantities refer to the plasma ion fluid.
The parameters νei and Z refer to the electron-ion collision frequency and ion
charge number, respectively. The gasdynamic pressure of the ion species is given
by Pi = nikBTi. A similar equation holds for the plasma electrons considered as
a fluid;
menedvedt
= neqE + neqve ×B−∇(nekBTe) +mneνei(vi − ve). (5.6)
If quasi-neutrality is assumed so that ne ≈ Zni, then the net velocity of the
ion and electron fluids is given by
v ≈ mivi + Zmvemi + Zm
and the average total pressure is
P = nikBTi + nekBTe ≈ ni(1 + Z)kBT
where the ions and electrons are assumed at the same temperature (alternatively,
without this assumption, simply define P ≈ nikB(Ti+ZTe)). The current density
is
J = q(neve − Znivi).
Using the above substitutions, adding Eqns (5.5) and (5.6) and ignoring terms
with me/mi << 1 gives [131] [9]
f = minidv
dt= −∇P + J×B + ε0
ω2p
ω2E · ∇E. (5.7)
106
5.2 The Nontransient Nonlinear Ponderomotive Force
Thus, the net force derived from a two-fluid description of a plasma yields
a nonlinear term in the electric field. The entire force expression given by Eqn
(5.7) will be referred to as the nonlinear ponderomotive force, although a more
general expression will be introduced shortly.
Adding the fluid equations for ions and electrons yielded the equation of mo-
tion in Eqn (5.7), subtracting Eqn (5.5) from Eqn (5.6), gives
m
q2ne
(dJ
dt+ νJ
)= E + v ×B +
1
qneJ×B +
1
qne
∇P1 + 1/Z
(see Appendix C of [18]) for a detailed derivation). Recalling Eqn (1.3), a gener-
alised Ohm’s Law for the plasma is found when the last three terms are neglected
(that is, the electric field is considered the most dominant contribution in a first
approximation);
dJ
dt+ νJ = ε0ω
2pE. (5.8)
The Lorentz theory of plasma is then described by a system of three equations
- two of Maxwell’s equations in vacuum and the integral of Eqn (5.8) where J
and E are assumed proportional to ei(k·r−ωt):
∇× E = −1
c
∂B
∂t
∇×B = µ0J +1
c2∂E
∂t
J = −ε0ω
2p
iω(1 + iν/ω)E.
In this way, the dynamics of laser-plasma interaction can be neatly described
by Maxwell’s microscopic equations (‘free space’) coupled with a particular form
of Ohm’s law that determines the specific physics for the case of a plasma.
While the previous discussion centred on the application of the two-fluid
model, an expression for the nonlinear ponderomotive force in a nondispersive
fluid dielectric was derived by Landau & Lifshitz [132] in terms of a stress tensor
such that the force density f was given as the divergence of the stress tensor σik
107
5.2 The Nontransient Nonlinear Ponderomotive Force
in the usual way;
fi =∂σik∂xk
.
Landau & Lifshitz themselves stated that the derivation of the ponderomotive
force for fluid dielectrics is rather complicated. Their derivation involved consid-
ering small variations in fluid position in the presence of a uniform electric field.
The specific form of the tensor derived by Landau & Lifshitz was
σik = −δikε0(P +
(εr − ρ
∂εr∂ρ
)E2
)+ ε0εrEiEk (5.9)
where P is the thermokinetic pressure, ρ the fluid density and εr the relative
permittivity of the fluid. In vacuum, εr = 1 and P = 0 and this tensor becomes
the Maxwell stress tensor from Eqn (2.5) in §2.2 (for zero magnetic field). Af-
ter taking the divergence of this tensor and doing some further manipulations,
Landau & Lifshitz showed that nonlinear ponderomotive force density is
f = −∇P + (ε− 1)∇E2 (5.10)
where ε = ε0εr is the total permittivity. Of course, a more general expression
including magnetic fields would include the time derivative of the Poynting vector
such that
f = ∇ · σ − ε0∂E×B
∂t.
It was Hora [9] who, motivated by Eqn (5.10), demonstrated the equivalence
of a generalised version of Schluter’s Eqn (5.7),
f = J×B + ε0E∇ · E + ε0∇ · (η2 − 1)E⊗ E, (5.11)
with
f = ∇ · (U + ε0(η2 − 1)E⊗ E)− ε0
∂E×B
∂t, (5.12)
where U is the Maxwell stress tensor in Eqn (2.5). A complete derivation of (5.12)
108
5.2 The Nontransient Nonlinear Ponderomotive Force
from (5.11 can be found in [18]. Note that the appearance of the permittivity in
Eqn (5.10) has been superseded by the refractive index in Eqn (5.12) in accordance
with Eqn (1.18) where η2 = εr when µr = 1. The pressure has also been neglected
as it is the nonlinear terms that are of real interest.
Hora showed that Eqn (5.12) not only contains Schluter’s ponderomotive term
from Eqn (5.7) but is also more general in that it includes spatial derivatives of the
refractive index [9] (required for inhomogeneous plasmas). It is remarkable that
Eqn (5.10), which was derived for nondispersive media, was successfully shown
to be applicable to plasmas with dispersion and absorption by Hora’s application
of the test of momentum conservation for an obliquely incident laser beam on a
stratified collisionless plasma [9].
After all the discussion of the complicated derivation of the ponderomotive
force by Landau and Lifshitz’s stress tensor method, or by Schluter’s two-fluid
method, this author notes that Eqn (5.12) is highly suggestive of a far simpler
method of deriving the nonlinear ponderomotive force density for a plasma in
an electromagnetic field. This simple method takes advantage of the fact that
it is already known that the Maxwell stress tensor U and time derivative of the
Poynting vector can be derived in the ‘canonical’ way from the Lagrangian
L =ε02E2 − 1
2µ0
B2,
as was discussed in §2.2. Presented with a problem in plasma physics, one could
be motivated to make a simple modification of the above Lagrangian by replacing
ε0 with ε0εr = ε0η2 to account for the relative permittivity of the plasma. Deriving
the energy-momentum tensor in the standard way and taking its divergence gives
a force density remarkably similar to that in Eqn (5.12):
f = ∇ ·(
U + ε0(η2 − 1)
(E⊗ E− 1
2IE2
))− ε0η2
∂E×B
∂t(5.13)
Only the fact that the refractive index appears as a coefficient for the IE2 and
109
5.3 The Transient Nonlinear Ponderomotive Force
Poynting terms ruins an exact match with Eqn (5.12). However, perhaps Eqn
(5.13) is more general than (5.12), as it would be expected that any alteration of
the permittivity to account for the plasma medium should affect every term in
the force expression that contains the electric field.
Both Eqn (5.12) and (5.13) reduce to the divergence of the Maxwell stress ten-
sor and Poynting time derivative when free space is considered (η = 1). All that
was required to derive Eqn (5.13) was a trivial alteration of the electromagnetic
Lagrangian to account for the change in permittivity of the electric field in the
presence of matter. No mention was made of plasmas and the result is completely
general until choosing to substitute a specific expression for the refractive index,
like one that incorporates plasma physics as in Eqn (1.17) where
η2 = 1−ω2p
ω(1− iν/ω).
The standard electromagnetic Lagrangian can easily be altered to include the
relative permeability of a plasma by substituting µ0µr for µ0 where µr 6= 1. That
such an elementary approach to the problem of laser-plasma physics should yield
results so similar to other (far more complicated) approaches can be viewed as
a validation of both these other approaches and the central tenet of this thesis -
that ‘canonical’ methods applied to complicated physics problems can still yield
interesting results. A canonical approach is highly attractive since all equations
of motion and notions of conserved quantities are exact and completely consistent
with each other, having all arisen from one seminal functional - the Lagrangian.
5.3 The Transient Nonlinear Ponderomotive Force
While Eqn (5.12) accounted for an inhomogeneous electric field (one that was not
constant over all space under consideration), it did not account for any change
in the ponderomotive force over time, as is required to describe the process of
simply switching on and off the laser beam. An additional transient expression
110
5.3 The Transient Nonlinear Ponderomotive Force
for laser-plasma interaction was explored by Zeidler et al. [133] and completed
by Hora in 1985 [12]. The original expression in 3-dimensional vector notation
for the time-dependent force is
f = ∇ ·(
U +
(1 +
1
ω
∂
∂t
)ε0(η
2 − 1)E⊗ E
)− ε0
∂
∂tE×B. (5.14)
Note that the extra terms added to the force expression by Hora are not
Lorentz invariant since it does not take into account the magnetic fields that would
appear outside the rest frame. Also, the argument of momentum conservation
could not be used to validate Eqn (5.14) as was the case with Eqn (5.12). In
an attempt to overcome this, the expression was generalised by Rowlands [11]
[134] to include magnetic fields by expressing the force using the electromagnetic
tensor, such that
f ν = T νµ,µ = Tνµ,µ +∂
∂xµ
((1 +
c
ω
∂
∂t
)ε0(η
2 − 1)
µ20σ
2F ντ
,τFµγ,γ
)(5.15)
where the time derivative of the Poynting vector is absorbed into the complete
4×4 stress-energy tensor Tνµ in the usual way. The conversion of Eqn (5.14) into
an expression that was both gauge and Lorentz invariant was seen as confirmation
of its correctness [18]. However, it is argued in this thesis that (5.15) is not the
correct generalisation of the transient nonlinear ponderomotive force (5.14) for
three separate reasons.
Firstly, the process used by Rowlands to transit from (5.14) to (5.15) involved
taking advantage of Ohm’s Law and Maxwell’s equations whereby
Ei =1
σJ i and − µ0J
µ = F µτ,τ ,
thus substituting a four-vector for the electric field as follows:
Ei → − 1
µ0σF µτ
,τ .
111
5.3 The Transient Nonlinear Ponderomotive Force
In this way, Hora’s expression was neatly converted to four-dimensional no-
tation while also bringing magnetic fields into the fold with the electromagnetic
tensor. The process of substituting Ohm’s Law into Ampere’s Law to describe
plasma physics is reasonable and was used in §1.2.4, §1.2.5 and §5.2 to derive
the dispersion relation. However, if all the relevant equations - Lorentz force,
Maxwell’s equations, Ohm’s Law - are to be physically consistent with the tran-
sient ponderomotive force, then they must all spring from the same source. This
author has already pointed out in §5.2 that while the non-transient ponderomo-
tive force appears to alter the energy-momentum tensor, it (or something very
close to it) can still be derived from the standard electromagnetic Lagrangian
with a simple modification, thus leaving everything else constructed from the La-
grangian (the form Maxwell’s equations) untouched in the process and ensuring
that the whole system is self-consistent. The tensor being differentiated in Eqn
(5.14) is suggestive of a higher derivative theory, a theory that would yield mod-
ifications to Maxwell’s equations at least, and yet the usual equations of motion
are assumed to be valid substitutions in transiting to Eqn (5.15).
While Rowlands was not concerned with higher-derivative Lagrangian theo-
ries, it is clear that in a consistent physical model, if the energy-momentum tensor
differs from the ‘canonical’ expression by the addition of higher-derivatives of the
field, there is no reason to expect the standard Euler-Lagrange equation,
−µ0Jµ = F µτ
,τ , (5.16)
to hold. For example, it has already been shown that in the case of the Podol-
sky Euler-Lagrange equation (4.18), if the Podolsky Lagrangian was modified to
include the standard interaction term JαAα for matter in the presence of the
electromagnetic field, then the Euler-Lagrange equation (4.18) would be
−µ0Jµ = (1 + µ0a
2 ∂2
∂xγxγ)F µτ
,τ .
112
5.3 The Transient Nonlinear Ponderomotive Force
Indeed, the Podolsky Lagrangian presents the biggest challenge to Eqn (5.15).
It was discussed in §4.3 that the Podolsky Lagrangian is the only Lorentz invariant
generalisation of the electromagnetic Lagrangian that yields equations of motion
below sixth-order, yet Eqn (5.15) introduces a term into the tensor that is one
order of derivative higher than anything in the Podolsky tensor (4.21) and only
shares one term (the nontransient term) with the Podolsky tensor out of the four
or five extra terms it contains (depending on whether the tensor in (4.21) or
(4.26) is considered). Since the Podolsky Lagrangian is well known to be the only
Lorentz invariant generalisation of electrodynamics yielding linear equations of
motion below sixth order, it would be expected that any other energy-momentum
tensor expression containing higher derivatives would be some special case of the
Podolsky tensor, but there appears to be no real correlation between the Rowlands
and Podolsky tensors.
Secondly, high temperature, high energy laser-plasma interactions (seen in
ICF, for example) often assume that the plasma is collisionless, in which case the
conductivity is extremely high or approximately infinite. The appearance of the
square of the conductivity in the denominator of the nonlinear ponderomotive
terms of Eqn (5.15) then seems to make them negligible or effectively zero, in
contrast to Hora’s assertion that the nonlinear ponderomotive force dominates
above thermal effects in certain regimes [18].
Thirdly, the transient nonlinear ponderomotive force in Eqn (5.14) is supposed
to account for the time variation in the field itself and it would be expected that
the dominant temporal contribution to the physics would come from the field
and perhaps its derivatives, if not just the field itself. However, only the time
derivative of the derivatives of the field appears in Eqn (5.15).
A final minor point regarding the way Rowlands argued for the gauge and
Lorentz invariance of Eqn (5.15). The expression is of course gauge invariant
since the electromagnetic tensor and its derivatives are gauge invariant. However,
Rowlands mentioned in two papers [11] [134] that the laser frequency ω is a
113
5.3 The Transient Nonlinear Ponderomotive Force
scalar and therefore Lorentz invariant. Of course, the frequency is not invariant
and would be blue- or red-shifted in any frame of reference moving respectively
toward or away from the laser source. The real reason that the appearance of the
laser frequency does not violate Lorentz invariance is the same reason that the
partial derivative with respect to time does not; they appear in the same factor,
1
ω
∂
∂t,
which is dimensionless and therefore invariant under any transformation of space-
time coordinates.
We present here an original alternate method of generalising Eqn (5.14) to four
dimensions while also ensuring gauge and Lorentz invariance. First, consider Eqn
(5.14) with the purely notational changes
Ei → cF i0 → cF µ0.
Note that the components of the electric field can be sent to components of
the electromagnetic tensor, without ambiguity, with a change in the range of the
index i = 1, 2, 3 to µ = 0, 1, 2, 3 since F 00 = 0. This transformation to a four-
vector was what Rowlands did not consider to be possible, and hence he sought
to transform the electric field to a four-vector via the current. Now Eqn (5.14)
looks like
fµ = Tµν,ν +∂
∂xν
((1 +
1
ω
∂
∂t
)η2 − 1
µ0
F µ0F ν0
). (5.17)
The energy-momentum tensor is then simply
T µν = Tµν +
(1 +
1
ω
∂
∂t
)η2 − 1
µ0
F µ0F ν0. (5.18)
If an inclusion of magnetic fields is sought, then perhaps the simplest alteration
would be to ‘allow the magnetic field in’ by changing the indices that are fixed
114
5.3 The Transient Nonlinear Ponderomotive Force
at 0 to dummy indices α such that
T µν = Tµν +
(1 +
1
ω
∂
∂t
)η2 − 1
µ0
F µαF να. (5.19)
When α = 0, there is no change of sign in lowering the index where F ν0 → F να,
so Eqn (5.19) is consistent with (5.18). When α 6= 0, it is true that F να =
−F να, but since the original expression did not include these terms, the change
of sign can be considered part of the generalisation. Now the ponderomotive
term in (5.19) again appears as a ‘refractive’ modification to the Maxwell energy-
momentum tensor Tµν , which contains the term
1
µ0
F µαF να.
This is analogous to the three dimensional result in Eqn (5.13), although
the time derivative of the ponderomotive term in (5.19) still escapes a canonical
explanation as it is one order of derivative lower than any of the Podolsky terms
in (4.21). However, (5.19) is Lorentz and gauge invariant.
Compare (5.19) to the Rowlands expression,
T νµ = Tνµ +
(1 +
c
ω
∂
∂x0
)ε0(η
2 − 1)
µ20σ
2F ντ
,τFµγ,γ. (5.20)
The difference between (5.19) and (5.20) is of course the higher derivatives
and squared conductivity term in the denominator of Rowlands’ expression. Is
(5.19) any more physically meaningful as a generalisation of the tensor in Eqn
(5.14)? This question probably requires experimental verification in laser-plasma
experiments at the kind of ultra high laser intensities where the ponderomotive
force becomes dominant. However, the advantage of (5.19) over the Rowlands
expression is that it did not require any assumptions about the field equations to
derive it, merely a change in notation and a ‘generalisation’ in letting the indices
of the electromagnetic tensor roam across their full range. The new expression
115
5.3 The Transient Nonlinear Ponderomotive Force
(5.19) also contains time derivatives of the field and not the field derivatives,
which seems more physically sensible and in line with the original Eqn (5.14).
The tensor in (5.19) is also, like the Rowlands expression, both gauge and Lorentz
invariant but still appears to lack any canonical way of deriving the transient term
from a Lagrangian.
It is this author’s opinion that the transient nonlinear ponderomotive force
derived from (5.19),
fµ = Tµν,ν +∂
∂xν
((1 +
1
ω
∂
∂t
)η2 − 1
µ0
F µαF να
), (5.21)
is the correct generalisation of Eqn (5.14), or at least it is the gauge and Lorentz
invariant version that most closely approximates Eqn (5.14).
116
6
Concluding Remarks
Closing the gap between theory and experiment in ICF schemes is likely going
to take many more years, or even decades. Every time a new and more powerful
laser system has been built to achieve ignition in ICF, the additional energy of
the laser system served to magnify other physical complexities and negate the
simulations based on earlier systems. It is currently not clear whether ICF will
ever be useful on a commercial scale. However, hope still remains in the fact
that physicists have not reached a dead end in what they do not know. Consider
the particle physicists at the Large Hadron Collider (LHC) who, having recently
found the Higgs boson exactly as they expected to, are left with no obvious leads
to follow (it was hoped that some deviation from predictions would light the way
to new physics.) The situation was elegantly summed up in a recent article which
stated that “NIF physicists wish their simulations were better; LHC physicists
complain that theirs were too good” [4].
The extreme condensed matter states encountered ICF are not well under-
stood and it is likely that all the principles and assumptions used in the past
are no longer good enough. Nature is always described by the laws of quantum
mechanics and relativity (and some unknown laws in between), whether or not we
can practically ignore them in everyday pursuits. The question of how much can
still be practically ignored in ultra-high intensity laser-plasma physics remains
open. Even taking into account the magnitude of complexity of these problems,
117
it may be that new physics is required to accurately describe laser-plasma inter-
action at high energies. Some have suggested that it is even possible to probe the
physics of particle pair production at energy levels currently available [135].
With these problems in mind, the main thesis put forward in the preced-
ing chapters is that analytical solutions to problems formulated in terms of La-
grangian or Hamiltonian physics can yield interesting results, even when applied
to fields as fraught with complexity as plasma physics. Much of the literature
reviewed herein has focused on reproducing equations of motion that are already
well-known to find some added level of insight through the canonical structure,
if not for purely aesthetic reasons.
This author submits that it is far more interesting to guess a new Lagrangian
or Hamiltonian for a given problem and then derive all the equations of motion
and conservation laws via Hamilton’s principle. The results may not coincide
with other expressions, indeed, it would be hoped that they do not if any new
insight is to be gained. All the salient features of the problem can be uncovered
provided the choice of Lagrangian functional is as reasonable a guess at the true
nature of the physics as possible.
It was this idea that led this author to propose in Chapter 4 that the Podolsky
Lagrangian could be applied to the case of laser-plasma interaction. The Podol-
sky Lagrangian represents a simple modification to Maxwell’s electrodynamics
that eliminates the infinite self-energy of the electron and incorporates higher
frequencies that are usually neglected in a first approximation. If the Podolsky
Lagrangian represents the true nature of electromagnetism with a new constant of
nature determining physics below the femtometre scale, then it is most definitely
applicable to the extreme condensed matter states encountered in ICF. However,
if the Podolsky Lagrangian does not represent the true physics of electromag-
netism, then perhaps its form could still be used to describe plasma physics at a
certain scale, specifically, that of the Debye length in a plasma. In either case,
it is this author’s opinion that the results presented in Chapter 4, especially the
118
new plasma dispersion relation Eqn (4.27), will be of use to physicists studying
laser-plasma interaction now and in the future.
As a side note to the general theme in Chapter 4, it is worth noting that this
author’s derivation of a gauge invariant energy-momentum tensor for any gauge
invariant Lagrangian dependent on second order derivatives of the coordinates is
fully general. Even if a particular Lagrangian is not gauge invariant, any gauge
invariant terms can be separated and the corresponding tensor found with ease
according to Eqn (4.15), which may simplify calculations considerably.
In Chapter 5, the case of the nonlinear ponderomotive force in laser-plasma
interactions was addressed. After reviewing a derivation of the nonlinear pondero-
motive force based on the two-fluid plasma model, and another derivation based
on a stress tensor method due to Landau & Lifshitz and Hora, it was pointed out
by this author that something very close to this force expression could be derived
via Hamilton’s principle. All that was required for this canonical derivation of
the force was a modification of the coefficient of the squared electric field mag-
nitude in the standard electromagnetic Lagrangian to account for the fact that
it describes a laser beam interacting with a dielectric (plasma). This method
was extended in a critical review by this author of the work by Rowlands, who
had derived what he considered to be the Lorentz invariant expression of the
transient nonlinear ponderomotive force in covariant notation. However, it was
pointed out by this author that a simpler generalisation of the transient nonlinear
ponderomotive force was possible, and that this generalisation was more in line
with the idea that the force is a special case of the usual canonical expression,
albeit modified for a plasma.
The results of this thesis have therefore flown entirely from the realm of vari-
ational principles applied to plasma physics. In one sense, it is no surprise to
experienced physicists that the equations of motion for any system should be de-
rived so easily from Hamilton’s principle, given an appropriate Lagrangian. On
the other hand, it seems quite surprising that such results should be found for
119
the fiendishly complicated field of laser-plasma interaction. Whether or not the
original material presented in this thesis will improve the accuracy of theoretical
models of laser-plasma interaction is a question that can only be answered by
experiments. However, the central tenet of this thesis remains that Hamilton’s
principle is the most powerful tool at the disposal of any theoretical physicist,
and that physicists should not be daunted by applying it to ‘real-world’ problems
that appear to be too complicated for it to handle.
120
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