Classical Mechanics (Kibble & Berkshire) · Problem 7.10

Non-linear oscillations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · nearly-circular-orbits

Problem

A particle of mass

m

moves under the central field

\boldsymbol{F} = -m\left(\frac{\gamma\, e^{-\epsilon r/a}}{r^2}\right)\hat{\boldsymbol{r}},

where

\gamma

a

and

\epsilon

are positive constants. Find the apsidal angle for a nearly circular orbit of radius

a

. When

\epsilon

is small, show that the perihelion of the orbit advances by approximately

\pi\epsilon

on each revolution.