A particle of mass moves in the trap H= p_x^2+p_y^2+p_z^2 2m + 12 ,(x^2+y^2)+ 12 _z z^2

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

Action and Symmetry, Ch Poisson brackets, phase flow and Liouville's theorem

Problem

A particle of mass

m

moves in the trap

H=\frac{p_x^2+p_y^2+p_z^2}{2m}+\tfrac12\kappa\,(x^2+y^2)+\tfrac12\kappa_z z^2,

with

\kappa=2.0\,\mathrm{N\,m^{-1}}

and

\kappa_z=8.0\,\mathrm{N\,m^{-1}}

, so the trap is symmetric about the

z

axis only. ((a)) Using the Leibniz rule and

\{L_i,q_j\}=\epsilon_{ijk}q_k

\{L_i,p_j\}=\epsilon_{ijk}p_k

, show that

\{L_z,H\}=0

and

\{L_x,H\}=(\kappa-\kappa_z)\,yz

, and find

\{L_y,H\}

in the same way. ((b)) At an instant when the particle is at

(x,y,z)=(0.10,\,0.20,\,0.30)\,\mathrm{m}

, evaluate

\{L_x,H\}

and

\{L_y,H\}

. Which components of

\mathbf L

are constants of the motion? ((c)) Someone proposes adding a further potential to this trap, chosen so that

L_x

becomes conserved while

L_z

stays conserved and

L_y

does not. Use Poisson's theorem and

\{L_z,L_x\}=L_y

to show that no such potential exists.

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