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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. - PowerPoint PPT Presentation

Antiproton Stacking and Cooling

Hamiltonian Formalism

Eric Prebys, FNALMotivationWe 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, its useful to at least become familiar with more formal Lagrangian/Hamiltonian based approachCan handle problems too complex for kinematic approachMore common in advanced textbooks and papersEventually intuitive

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism2Review*The Lagrangian of a body is defined as

Hamiltons variational principle says that the body will follow a trajectory in time (or other independent variable) which minimizes the action

Generalized forceUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism3*Nice treatment in Reiser, Theory and Design of Charged Particle Beams

Potential energy Kinetic Energy

Demonstration in Cartesian CoordinatesLagrangian

Equations of motion

In other words

Lagrangian mechanics is really just a turnkey way to do energy conservation in arbitrary coordinate systems.USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism4

E&MIntroduce velocity-dependent force: Lagranges equations still hold for

We describe the magnetic field in terms of the vector potential

The Lorentz force now becomes, eg

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism5

Lorentz Gauge

HomeworkRelativistic VersionWe want to find a relativistically correct Lagrangian. Assume for now

In Cartesian coordinates, we have eg.

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism6

Make the substitution

Check in Cartesian coordinates for B=0

More generallyUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism7

Canonical MomentumLagranges 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 coordinatesUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism8

canonical momentumordinary momentumHamiltons EquationsIntroduce HamiltonianWe take the total differential of both sides

Equating the LHS and RHS gives us

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism9

LHSRHS

HamiltonsEquations of motionConservation lawsFrom the last equation, we have

In other words, the Hamiltonian is conserved if there is no explicit time dependence of the Lagrangian.USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism10

Particle in an Electromagnetic FieldRecall

In Cartesian coordinatesUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism11

Total EnergyHamiltonian in Canonical MomentumIn order to apply Hamilton equations, we must express the Hamiltonian in terms of canonical, rather than mechanical momentumUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism12

Remember this forever!Change of Coordinates and Generator FunctionsWe 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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism13

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

Lets 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 termsUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism14

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:USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism15

solve for p and P in terms of q and QHamiltonian in terms of new variables

In all cases

Example: Harmonic OscillatorUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism16We know the Hamiltonian is

and

change variables to

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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism17So we have

J has units of Energy*time actionPhase angleThese 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 SystemUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism18Assume 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 constantGeneral caseUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism19

We start with a known systemWe transform to a system which represents small deviations from this system

Use a generating function of the second type

integrateUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism20We can calculate the new Hamiltonian and expand for small deviations about the equilibrium

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

Its 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 RevisitedUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism21Recall we showed that

Canonical momentum!We recall our coordinate system from an earlier lecture

Reference trajectoryParticle trajectoryAnd define canonical s momentum and vector potential as

Use new symbolUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism22We 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) thatUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism23Consider 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 momentumFor small deviations

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism24We showed that the first few terms of the magnetic field are

dipolequadrupolesextupoleWe have

You can show (homework) that this is given by

We have

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism25In the case where we have only vertical fields, this becomes

Normalize by the design momentum

At the nominal momentum =0, so

same answer we got beforeUSPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism26By comparing this to the harmonic oscillator, we can write

We have a solution of the form

Look for action-angle variables

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 10 - Hamiltonian Formalism27Look for a generating function such that

Integrate to get

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