Hamiltonian Formalism

Hamiltonian Formalism

Eric Prebys, FNAL

Motivation We have focused largely on a kinematics based

approach to beam dynamics. Most people find it more intuitive, at least when first

learning the material. However, it’s useful to at least become familiar

with more formal Lagrangian/Hamiltonian based approach Can handle problems too complex for kinematic approach More common in advanced textbooks and papers Eventually intuitive

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism 2

Review* The Lagrangian of a body is defined as

Hamilton’s variational principle says that the body will follow a trajectory in time (or other independent variable) which minimizes the “action”

Generalized force


*Nice treatment in Reiser, “Theory and Design of Charged Particle Beams”

Potential energy Kinetic

Energy

Demonstration in Cartesian Coordinates Lagrangian

Equations of motion

In other words

Lagrangian mechanics is really just a turnkey way to do energy conservation in arbitrary coordinate systems.


E&M Introduce velocity-dependent force: Lagrange’s equations still hold for

We describe the magnetic field in terms of the vector potential

The Lorentz force now becomes, eg


Lorentz Gauge

Homework

Relativistic Version We want to find a relativistically correct

Lagrangian. Assume for now

In Cartesian coordinates, we have eg.


Make the substitution

Check in Cartesian coordinates for B=0

More generally


Canonical Momentum Lagrange’s equations are second order diff. eq.

We will find that it will be useful to specify system in term of twice as many first order diff. eqs.

We introduce the “conjugate” or “canonical” momentum

In Cartesian coordinates


canonical momentum

ordinary momentum

Hamilton’s Equations Introduce “Hamiltonian” We take the total differential of both sides

Equating the LHS and RHS gives us


LHS

RHS

Hamilton’sEquations of motion

Conservation laws From the last equation, we have

In other words, the Hamiltonian is conserved if there is no explicit time dependence of the Lagrangian.


Particle in an Electromagnetic Field Recall

In Cartesian coordinates


Total Energy

Hamiltonian in Canonical Momentum In order to apply Hamilton’ equations, we must

express the Hamiltonian in terms of canonical, rather than mechanical momentum


Remember this forever!

Change of Coordinates and Generator Functions We will often find it useful to express the

Hamiltonian in other coordinate systems, and need a turnkey way to generate canonical coordinate/momentum pairs. That is

We construct the Lagrangian out of the new coordinates

We still want the action principle to hold


This means that the new and old Lagrangians can differ by at most a total time derivative

Let’s first consider a function which depends only on the new and old coordinates

Then we must have

Expand the total time time derivative at the right and combine terms


Because q and Q are independent variables, the coefficients must vanish.

F1 is called the “generating function of the canonical transformation. Rather than choosing (q,Q) as variables, we could have chosen (q,P), (Q,p) or (p,P). The convention is:


solve for p and P in terms of q and Q

Hamiltonian in terms of new variables

In all cases

Example: Harmonic Oscillator


We know the Hamiltonian is

andchange variables to

we want the old momentum in terms of the new and old coordinate


So we have

J has units of Energy*time “action”

Phase angle

These are known as “action-angle” variables. We will see that this will be very useful for studying systems which are perturbed by the addition of small non-linear terms.

Deviations from a Periodic System


Assume we have a system with solutions x0 and y0, which are periodic with period T

Now consider an orbit near the periodic orbit

Substituting in and expanding, we get

These are the equations one obtains with a Hamiltonian of the form (homework)

periodic(!) in time rather than constant

General case


We start with a known system

We transform to a system which represents small deviations from this system

Use a generating function of the second type

integrate


We can calculate the new Hamiltonian and expand for small deviations about the equilibrium

No dependence on Q or P, so can be ignored!

It’s important to remember that these coefficients are derivatives of the Hamiltonian evaluated at the unperturbed orbit, so in general they are periodic, but not constant in time!

Particle Motion Revisited


Recall we showed that

Canonical momentum!

We recall our coordinate system from an earlier lecturex

ys

xr

Reference trajectory

Particle trajectory

And define canonical s momentum and vector potential as

Use new symbol


We would like to change our independent variable from t to s. Note

We can transform this into a partial derivative by setting the total derivative to zero. In general

so

new Hamiltonian

You can show (homework) that


Consider a system with no E fields and only B fields in the transverse directions, so there is only an s component to the vector potential

In this case, H is the total energy, so

normal “kinetic” momentum

For small deviations


We showed that the first few terms of the magnetic field are

...2

~~~

...~2

~

220

220

xyByxByBxBBB

xyByxByBxBBB

x

y

dipole

quadrupole

sextupole

We have

You can show (homework) that this is given by

We have


In the case where we have only vertical fields, this becomes

Normalize by the design momentum

At the nominal momentum ρ=ρ0, so

same answer we got before


By comparing this to the harmonic oscillator, we can write

We have a solution of the form

Look for action-angle variables


Look for a generating function such that

Integrate to get

In an analogy to the harmonic oscillator, the unperturbed Hamiltonian is

Hamiltonian Formalism

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