A two-dimensional harmonic trap has a stiffness that is deliberately ramped in time…

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 two-dimensional harmonic trap has a stiffness that is deliberately ramped in time (a "breathing trap"), with Hamiltonian H(q1,q2,p1,p2,t)=p12+p222m+12m ω(t)2 (q12+q22),H(q_1,q_2,p_1,p_2,t) = \frac{p_1^2+p_2^2}{2m} + \frac12 m\,\omega(t)^2\,(q_1^2+q_2^2), where mm is a constant mass and ω(t)\omega(t) is an arbitrary smooth, strictly positive function of time. A large ensemble of such traps is described by a phase-space density ρ(q1,q2,p1,p2,t)\rho(q_1,q_2,p_1,p_2,t). ((a)) Show that HH is not a constant of the motion when ω˙≠0\dot\omega\neq0, but that L=q1p2−q2p1L=q_1p_2-q_2p_1 is conserved for every choice of ω(t)\omega(t). ((b)) A density that follows the instantaneous energy shells, ρ(z,t)=g(H(z,t))\rho(\mathbf z,t)=g\big(H(\mathbf z,t)\big) for some function gg, looks like the natural extension of the oscillator's stationary ring. Using the Liouville equation, show that it is not a solution at any instant when ω˙≠0\dot\omega\neq0, wherever g′≠0g'\neq0. ((c)) Show that every density ρ=F(L)\rho=F(L), with no explicit time dependence, satisfies the Liouville equation for every ω(t)\omega(t) and so is stationary. Then explain why no such density, other than zero, can describe a finite ensemble, one for which ∫ρ dq1 dq2 dp1 dp2\int\rho\,\mathrm{d}q_1\,\mathrm{d}q_2\,\mathrm{d}p_1\,\mathrm{d}p_2 is finite.

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