A cloud of free particles of mass

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

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

Problem

m=0.50\,\mathrm{kg}

, each with no force acting on it, occupies the square

0\le x\le 2.0\,\mathrm{m}

0\le p\le 2.0\,\mathrm{kg\,m\,s^{-1}}

in the

(x,p)

plane at

t=0

. After

t=4.0\,\mathrm{s}

the square has sheared into a parallelogram, since a particle with momentum

p

moves a distance

pt/m

. Someone argues: "The parallelogram stretches from

x=0

out to

x=18\,\mathrm{m}

, an

x

-extent of

18\,\mathrm{m}

, far bigger than the original

2.0\,\mathrm{m}

. Liouville's theorem says the phase-space area is conserved at

4.0\,\mathrm{J\,s}

, so the momentum spread must have shrunk to compensate: the new momentum spread is

4.0\,\mathrm{J\,s}/18\,\mathrm{m}=0.22\,\mathrm{kg\,m\,s^{-1}}

." ((a)) Find the error in this argument. ((b)) State the true momentum spread at

t=4.0\,\mathrm{s}

and verify that the parallelogram's actual area is still

4.0\,\mathrm{J\,s}

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