The reduced Kepler problem (Chapter 8) has Hamiltonian

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

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

Problem

The reduced Kepler problem (Chapter 8) has Hamiltonian, in the phase-space variables of Chapter 13,

H(r,\varphi,p_r,p_\varphi)=\frac{p_r^2}{2\mu}+\frac{p_\varphi^2}{2\mu r^2}-\frac{k}{r},

with reduced mass

\mu=2.0\,\mathrm{kg}

and

k=5.0\,\mathrm{N\,m^2}

. At some instant the orbit has

r=1.0\,\mathrm{m}

p_r=0.50\,\mathrm{kg\,m\,s^{-1}}

and

p_\varphi=3.0\,\mathrm{kg\,m^2\,s^{-1}}

. ((a)) Compute

\{p_\varphi,H\}

and use the bracket test to say whether

p_\varphi

is conserved, connecting the result to why

\varphi

does not appear in

H

. ((b)) Compute

\{p_r,H\}

at the stated instant and say whether

p_r

is conserved.

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