Classical Mechanics (Kibble & Berkshire) · Problem 7.2

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

Physics — Mechanics · radial-motion-equation-apses

Problem

A particle

P

of mass

m

moves under the attractive inverse square field

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

. Initially

P

is at a point

C

, a distance

c

from

O

, when it is projected with speed

(\gamma/c)^{1/2}

in a direction making an acute angle

\alpha

with the line

OC

. Find the apsidal distances in the resulting orbit. Given that the orbit is an ellipse with

O

at a focus, find the semi-major and semi-minor axes of this ellipse.