Classical Mechanics (Kibble & Berkshire) · Problem 7.9

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

Physics — Mechanics · path-equation

Problem

A particle of mass

m

moves under the central field

\boldsymbol{F} = -m\gamma^2\left(\frac{4}{r^3} + \frac{a^2}{r^5}\right)\hat{\boldsymbol{r}},

where

\gamma

and

a

are positive constants. Initially the particle is at a distance

a

from the centre of force and is projected at right angles to the radius vector with speed

3\gamma/\sqrt{2}\,a

. Find the polar equation of the resulting path and make a sketch of it. Find the time taken for the particle to reach the centre of force.