For planar Kepler motion in the - plane

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

Action and Symmetry, Ch Hamilton-Jacobi theory and action-angle variables

Problem

For planar Kepler motion in the

x

y

plane, with

r=\sqrt{x^2+y^2}

, the angular momentum component is

\ell=xp_y-yp_x

, and the Runge-Lenz vector's in-plane components are

A_x=\ell p_y-\mu k\frac{x}{r},\qquad A_y=-\ell p_x-\mu k\frac{y}{r},

with

\mu

the reduced mass and

k

the strength of the attractive potential

V=-k/r

. Using the canonical brackets

\{x,p_x\}=\{y,p_y\}=1

, all others zero, and the Leibniz rule for Poisson brackets,

\{f,gh\}=\{f,g\}h+g\{f,h\}

: ((a)) Compute

\{x,A_x\}

and

\{x,A_y\}

. ((b)) Explain why the result shows that the change of

x

generated by

A_x

A_y

cannot be produced by any transformation of the coordinates alone.

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