Classical Mechanics (Kibble & Berkshire) · Problem 7.3

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

Physics — Mechanics · radial-motion-equation-apses

Problem

A particle of mass

m

moves under the attractive inverse square field

\boldsymbol{F} = -(m\gamma/r^2)\hat{\boldsymbol{r}}

. Show that the equation satisfied by the apsidal distances is

2Er^2 + 2\gamma r - L^2 = 0,

where

E

and

L

are the specific total energy and angular momentum of the particle. When

E < 0

, the orbit is known to be an ellipse with

O

as a focus. By considering the sum and product of the roots of the above equation, establish the elliptic orbit formulae

L^2 = \gamma b^2/a, \qquad E = -\gamma/2a.