Classical Mechanics (Kibble & Berkshire) · Problem 7.8

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

Physics — Mechanics · path-equation

Problem

A particle

P

of mass

m

moves under the central field

\boldsymbol{F} = -(m\gamma^2/r^5)\hat{\boldsymbol{r}}

, where

\gamma

is a positive constant. Initially

P

is at a great distance from

O

and is projected towards

O

with speed

\sqrt{2}\gamma/p^2

along a line whose perpendicular distance from

O

p

. Show that the polar equation of the path of

P

is given by

r = \frac{p}{\sqrt{2}}\coth\!\left(\frac{\theta}{\sqrt{2}}\right).

Make a sketch of the path.