A particle travels along a beamline in the transverse coordinate (Chapter 2) with…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Canonical transformations, generating functions and symplectic structure

Problem

A particle travels along a beamline in the transverse coordinate

x

(Chapter 2) with canonical momentum

p_x

(Chapter 13, the Legendre transform of the Lagrangian in the transverse velocity). It passes through a thin focusing element that leaves

x

unchanged but changes the canonical momentum by

\Delta p_x=-\kappa x

, where

\kappa

is the lens strength. This kick is generated by

F_2(x,P_x)=xP_x+\tfrac12\kappa x^2.

((a)) Derive

p_x

and

X

(the position after the lens) from

F_2

, and hence write the new momentum

P_x

after the lens in terms of

x

and

p_x

. Evaluate

X

and

P_x

for

x=0.030\,\mathrm{m}

p_x=2.0\,\mathrm{kg\,m\,s^{-1}}

and

\kappa=5.0\,\mathrm{kg\,s^{-1}}

. ((b)) Using the Poisson bracket of Chapter 14 in the old variables

(x,p_x)

, compute

\{X,P_x\}

and state whether the kick is a canonical transformation.

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