Classical Dynamics of Particles and Systems · Problem 8-20

Dynamics of a System of Particles — Marion & Thornton

Physics — Mechanics · central-force-time-average-elliptical-orbit

Problem

For a particle moving in an elliptical orbit with semimajor axis

a

and eccentricity

\varepsilon

, show that

\langle (a/r)^4 \cos\theta \rangle = \varepsilon/(1-\varepsilon^2)^{5/2}

where the angular brackets denote a time average over one complete period.