APPLICATION OF HAMILTON-JACOBI THEORY TO VLASOV'S EQUATION
Transcript of APPLICATION OF HAMILTON-JACOBI THEORY TO VLASOV'S EQUATION
IC/66/95
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL
PHYSICS
APPLICATIONOF HAMILTON-JACOBI THEORY
TO VLASOV'S EQUATION
D. PFIRSCH
1966
PIAZZA OBERDAN
TRIESTE
IC/66/95
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
APPLICATION OF HAMILTON- JACOBI THEORY
TO VLASOV'S EQUATION*
D- PFIRSCH**
TRIESTE
July 1966
* To be submitted to Nuclear Fusion
** Permanent address: Max-Planck-Institut fur Physik und AstrophysUi, Munich, Fed. Rep. Germany
ABSTRACT
Solutions of Vlasov's equation are given in terms of the
Hamilton-Jacob! function S for the characteristics. Such
solutions are useful in order to derive equations for
macroscopic quantities which are of interest for instance
in turbulence theories.
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APPLICATION OP HAMILTON-JACOBI THEORY TO VLASOV'S EQUATION
One Important class of problems in plasma physics deals
with the description of plasma turbulence. Because there exist
many microinstabilities which can be the cause for turbulence,
a microscopic treatment is often necessary. The fundamental
equation is in this case Vlasov's equation, which holds if
collisions can be neglected. One can then distinguish 4 problems:
1.) One has first to derive for some macroscopic quantities,
like the fluctuating electric or magnetic fields, nonlinear
equations of the following kind: if C represents the macroscopic
quantities and A,B,D matrices of second, third and fourth rank,
then an equation
A- C t 3:C C f J>; CCC
has to be derived from Vlasov's equation and Maxwell's
equations by some kind of nonsecular perturbation theory
(see for instance (1))
2.) One has to introduce statistical concepts leading to hierarchy
equations for the quantities
where the brackets indicate ensemble averages. One must add
to this a closure prescription, for instance neglect of 4th
order correlations.
J5.) One has to apply asymptotic methods to these equations in
order to obtain kinetic wave equations.
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4.) One has to solve these kinetic wave equations.
In this paper I will be concerned with the first of these
points. It is well known that the classical Kami1ton-Jabobi-
theory allows in a straight forwardway a nonsecular perturbation
theory which is similar to perturbation theories in quantum
theory. An example will illustrate the differences between usual
perturbation theory and perturbation theory of the Hamilton-
Jacobi-equation:
The exact solution of the equation
x = (ft t? xwith X(o) * 0 , K(0) * *>C
is
Perturbation theory with the equation of motion yields to first
order in f
X & cc sink t t €°t (t **M- *''»* 0 ,
which expression contains a so called secular term, which restricts
the validity of this solution to times / « ;F . The Hamilton-
Jacob!-theory yields in first order
a; c<L(l-t)St'nh-r~ •This solution is valid for £« / and t «fx ,i.e.; for much
longer times than the first approximate solution. But as can be
seen from the exact first oder Hamilton Jacobi-expression, one
must develop in general a very difficult elimination procedure.
It is an essential point, that in applying Hamilton-Jacobi's
Theory to Vlasov's equation in order to obtain macroscopic
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quantities one is not concerned with any elimination problem.
There Is a second essential point shown in this paper namely
that the Hamilton-Jacobi method allows a rather easy and
straightforward nonsecular perturbation theory in such a way
that in order to obtain the equations for the macroscopic
quantities one need not solve equations for the distribution
function as in quasilinear or similar theories (see for instance
(1) and the literature cited there). To gain this advantage,
however, one must sacrifice knowledge about microscopic quantities,
i.e. the distribution function in x,v - space.
1. Solution of Vlasov's Equation in terms of the Hamilton-
Jacobi Function
The Hamilton-Jacobi-equation of a Hamiltonian System with the
Hamiltonian H (£, xt t) is
In the following H is assumed to be the Hamiltonian in Vlasov's
equation
where the brackets indicate Polsson-brackets. A solution of (l)
can be written in the form
+ < . (3)
U, «£are constants of the integration; o& can be a vector with
as many components as X has* o^o can and will be dropped. £ has
the meaning of a generating function for a canonical trans-
formation. tL can be thought of as the new momenta and
2f (4)
as being the new coordinates, whereas the old momenta are
given by fo f&X . The Hamilton-Jacobi-equation expresses the
fact that the new Hamiltonian is identically equal to zero,
thus $£ and p, are constants of the motion.
Since any solution of Vlasov's equation can be written as a
function of the constants of motion, a solution of (2) is
f - / tyS).We are, however, not interested in such a function depending
on the variables <£ and P , because we wish later to calculate
macroscopic quantities which are functions of K and /
Therefore we introduce instead of |L the variable £ , which
can be done by relation (4). This transformation?however, is
not a canonical one. Thus in order to obtain the particle density
i n^ȣ -space we have to multiply (5) still by the Jacobian of
this transformation. This new function is therefore
(6)
With this function one calculates for instance the particle
density
or the current density (in case of a vanishing vector potential
and with e=m=l)
Similar we can find any other macroscopic quantity without per
forming any eliminations.
Of course if we wanted to know the particle density in x,£ -
space we would have to eliminate vi by use of the relation
A * at * w h i c h w o u l d D e l n general very difficult.
From (6) we may derive an interesting result as to the meaning
of the Haml1ton-Jacobi-Function in the framework of Vlasov's
equation: One take as the function f
(9)
so that
Thus, this Jacobian is just the probability to find a particle
at point 2L a t time t if its orbits are characterised by the
constant of motion *c 9 ^ . This corresponds closely to the
meaning of tP (ift'clg)^!}^ ̂ 1 In quantum theory, where in this case
cC0 are quantum numbers. A simple example is given by the
one dimensional motion of a particle in a time dependent potential,
The Hamilton-Jacobi equation for this problem is
* FmE * Fm
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We can solve this equation in the form
Vflt - - -t «*and
From this we calculate the Jacobian
(12)
2* fix \rM\where T/'(X,oC\ is the velocity of the particle characterised by
the constant of motion oC . It is well known that \/lit(*>*Q\ is Just
the prabability about which we have spoken.
To conclude this section I will give an example for the function
f. The example is a particle in a time varying but spatially
constant field of force g (t) say with g (t<to) = o, t0 finite.
Putting the mass m = 1 we have the Hamilton-Jacobi-eauation
ii i( X = 0 .(14)
A solution of this equation can be found in the form
(15)
with
4(0and
0
\ dr <j (r)"OO
(16)
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& I 4 (18)
(20)
From this we obtain according to (6)
or if we are interested in a spatially homogeneous solution:
(22)
It is remarkable that this function is exactly time-independent,
We will have to do later with similar features.
Clearly (22) yields a time-independent density, whereas the
current density is given by
(W \fjffj
In the usual V- representation we would have instead of (22)
(24)
eliminating 06 we obtain a time-dependent function. This
fact will also be of importance below.
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2. Perturbation theory
Clearly it is not possible in general to solve the Hamilton-
Jacobi-equation exactly. But as mentioned in the introduction
and as was illustrated by an example, the Hamilton-Jacobi-theory
allows a rather easy and straight-forward nonsecular perturbation
theory, which will be presented here.
Assume the Hamiltonian H can be decomposed into two parts
where Ho describes a problem which can be solved exactly, where-
as H, is a small perturbation. This means we assume to have
solved the equation
in the form
X = X (*,*,
by which the gi 5 are defined.
If we make the ansatz
(28)
where S,, S2j'.are higher and higher order contributions to S in
the sense of the smallness of H,, then we obtain for these
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quantities equations of the form
9s, u f K
- ' ' *~ . (30)
and in general
Of Jtf OfZzH / ^f ZgJL - expression containing (So, ..i
S ) -e n i/ ^f. ZgJL - expression containing (So, ..i,
there M/fa ? *%fin '" indicate 0'%(fl, Xj/)fJfl\ *
All these equations can be solved by integrating along the
characteristics, which are given by
M
These equations are just the equations for the unperturbed
paths of the particles characterised by the constants o£ .
These paths can be described by a time parameter Z~ as
From this it follows for instance
•-0Q
/f / ^ ' (2*)
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The lower limit ~ot> implies in general an adiabatic turning on
of the perturbation. For all that however, it can happen that
the integral diverges at the lower limit, because it could have
a nonvanishing meanvalue for /-*-<*> .In this case one has to
proceed in the following way:
We first decompose S, or in general S into two parts
- rOJ ,(35)
(2)
/7+J ft
=• - *n (36)
(2)where we define S v ' by the equation
is defined in eq. (31), for instance e--= Hj. Thus
n )
where the integral runs along the particle path I1 = 1 (x1)
Inserting this into equation (31) we obtain the equation
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Now it is possible in general to solve this equation in the
form
(39) •
where we will not encounter secularities.
J>. Application to the one-dimensional one-component plasma
The simplest case that one can treat is a one-dimensional one-
component plasma which is spatially homogeneous in the unper-
turbed state. Putting m = e=l we have as zero order equation
it
with the solution
C = - I
From this i t follows
Thus
is the exact form of the distribution f describing a plasma
which is homogeneous in the unperturbed state. I want to
emphasize again that f (<*) is exactly independent of time.
All the time-dependence and also all the x-dependence in a
perturbed state, is given by the Jacobian. This is similar to
the behaviour of the distribution function for the example in
section 1. Eliminating o£ by virtue of V^
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could however, obtain a time and space dependent function
The quantities we are interested in are the particle
density and the current density?00
— oo
i (45)
Putting the velocity of light equal to one, we can write the• •
Maxwellian equation JO e - r^V as
£- *4 \ Str [&;*«• <«>Thus in order to obtain for E an equation of third degree we
have to calculate [ %/}to third order in E, which can be done
by the perturbation theory described in section 2. In the present
context this perturbation theory is valid if the following two
conditions are fulfilled
(A) The velocity of a thermal particle must not be changed
appreciably during one period of the electric field,
i.e.
in 4/ ™
or equivalents, J£ *, I mih} tt 7JT
(B) The time needed by a particle travelling with the phase
velocity to move through a wave packet must hr small
compared to the oscillation period of a partiexe trapped
by the potential of the wave packet. If //£-# is
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the width of the wave packet and jf̂ s=
is the amplitude of the electric field then we must
have
4^ 0 ^ \ *r9&
or, equivalently,
Fir / ^ / /
The case in which the following third condition is also satis
fied, is often of interest:
(C) The inverse of the growthrate >*" is small compared
to the oscillation period of a trapped particle,i.e.
1
or, equivalently,
FT & 4-m ̂
Now performing the perturbation theory, we first write equation
(46) to third order:
Before determining all these expressions I will say something
about the second order and third order term. In our perturbation
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theory in section 2 equation (36) reads for n=2
IT (48)
Inserting this relation into the second order term in (47) we
are left with
instead of
is easily calculated from (38):
(*9)
Similarly we have for n = 3 from (36)
( 5 0 )
Thus we have again only to consider oCfSj /*%* which is similar
to (49) given by
06 W* W\T» Tx)\±
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In this expression 0->i IJX is the fall expression, being the
sum of (48) and (49).
We can now write down very easily the interesting quantities
of first, second and third order
CK>
(52)
?(53)
Wi fa
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3.1. Fourier transformation with respect to space and time
With regard to weak turbulence and some kinds of strong turbulence
it is useful to Fourier transform E (x,t) with respect to both
space and time:
Here we have introduced a periodicity length L by which the
TV
wavevectorsK are given by K = n -j- , n = O , ± l , ....* We can
then write e.g. (47) in terms of the first, second and third
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order contributions j ^ J2, j , to the current density. With
(52) to (54) we find easily
fa-Q-U } '(56)
oo
/ (58)
It is of interest to observe in which way secular terms are
avoided in these expressions. Secular terms would show up here
as terms leading to singularities at CJ = o. Because of the
left hand side of equation (47) which after Fourier transformation
is - iCJ E K (03) t such singularities would be given by terms in
the •ji's not vanishing for 03= o. Since for K = 0 the left hand
side is equal to zero also the right hand side vanishes identi-
cally In 0); then looking only for K=0 we find j., which is just
the usual first order current density, is exactly zero for £J=o.
The same is obviously true for the quantities j^ and j , because of
the factors CO in front.
I want to point out the fact that (56) to (58) are the exact
Fourier-transforms. There is no slow time-dependence left in
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&s in the qualilinear theory and therefore is also
no need to solve an additional equation for some quantity like
f (oC) ; f (cC) is here given exactly by the initial unperturbed
state of the plasma for which aC n a s the meaning of the velocity
of the particles. For later times oC is in general a very
complicated function of x,v,t, which however, we need not know.
Another point of interest is the fact that (T, and CTi+ )
can be expressed by linear combinations of o~ • This means
again a simplification over current theories.
Using (56) to (58) one can do now the weak turbulence theory as
described for instance in Kadomtsev's book (1), chapter II, or
the strong turbulence theory in the sense of Kraichnan (2), or
in a slightly modified way as also described in Kadomtsev's
book chapter III.
4. Conclusions
Through application of the Hamilton Jacobi-theory to the
Vlasov equation it was possible to find solutions which are
appropriate for calculating macroscopic quantities. For this
purpose one need not develop an elimination procedure of the
sort necessary for finding the motion of a particle; any
such procedure would generally be very difficult. In regard
to turbulence theory, for instance, it is an interesting
feature that this kind of formalism leads to a simple, straight
forward nonsecular perturbation theory, by which one easily finds
results not yet obtained with present theories. However, one
gains this advantage only by sacrificing knowledge about the
distribution function in x,v- space. On the other hand,
+) This was pointed out to me by Dr. Janussis.
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•,-. „ ;
at least for the case of a plasma which is homogeneous in the
unperturbed state, there exists a similar method for obtaining
an expression for the homogeneous part of the distribution function,
This method consists in solving the Vlasov equation by means of
the following equation:
where S is a function
and tfSfJLa.YQ constants of the motion, and x is given by
Preliminary results of this theory are similar to result*
obtained with the quasi-linear theory.
ACKNOWLEDGEMENTS
The author thanks Prof. Abdus Salam and the IAEA for their
hospitality at the ICTP, Trieste.
This work was sponsored by the Max-Planck-Institut fUr Physik
und Astrophysik, Munich, Germany.
REFERENCES
(1) B.B. Kadomtsev, Plasma Turbulence, Academic Press. London.New York (1965)
(2) R.H. Kraichnan, J. Fluid Mech. 5, 497 (1959)
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