Classical Mechanics (Kibble & Berkshire) · Problem 6.6

Orbits in a central field — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · oscillations-potential-energy

Problem

A particle

P

of mass

m

moves on the

x

-axis under the force field with potential energy

V = V_0(x/b)^4

, where

V_0

and

b

are positive constants. Show that any motion of

P

consists of a periodic oscillation with centre at the origin. Show further that, when the oscillation has amplitude

a

, the period

\tau

is given by

\tau = 2\sqrt{2}\left(\frac{m}{V_0}\right)^{1/2}\frac{b^2}{a}\int_0^1 \frac{d\xi}{(1-\xi^4)^{1/2}}.

[Thus, the larger the amplitude, the shorter the period!]