A planet of reduced mass orbits the Sun under with (Chapter 8's reduction of the…

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

Action and Symmetry, Ch The Hamiltonian, phase space and the variational principle in phase space

Problem

A planet of reduced mass

\mu=6.0\times10^{24}\,\mathrm{kg}

orbits the Sun under

V(r)=-k/r

with

k=8.0\times10^{44}\,\mathrm{J\,m}

(Chapter 8's reduction of the two-body problem). At pericentre the planet's distance is

r_{\rm p}=1.5\times10^{11}\,\mathrm{m}

and its speed (entirely tangential there) is

v_{\rm p}=3.0\times10^{4}\,\mathrm{m\,s^{-1}}

. ((a)) Starting from Chapter 8's planar Lagrangian

L=\tfrac12\mu(\dot r^2+r^2\dot\varphi^2)+k/r

, form the Routhian in the cyclic angle

\varphi

and write the resulting one-dimensional Hamiltonian

H_r=p_r^2/2\mu+V_{\rm eff}(r)

, in terms of the conserved angular momentum

\ell

. ((b)) Using

\ell=\mu r_{\rm p}v_{\rm p}

, evaluate

H_r

at pericentre, and check it agrees with

\tfrac12\mu v_{\rm p}^2-k/r_{\rm p}

computed directly. ((c)) State whether

H_r

is conserved and whether it equals the mechanical energy

T+V

, giving the reason from Chapter 4's criterion, and give its numerical value.

Browse the problem library