Classical Mechanics (Kibble & Berkshire) · Problem 8.3

The energy principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · lindstedt-method-unsymmetrical-oscillations

Problem

Unsymmetrical oscillations A non-linear oscillator satisfies the equation

\ddot{x} + x + \epsilon x^2 = 0,

where

\epsilon

is a small parameter. Explain why the oscillations are unsymmetrical about

x = 0

in this problem. Use Lindstedt's method to obtain a two-term approximation to

x(t)

for the oscillation in which the maximum value of

x

is unity. Deduce a two-term approximation to the minimum value achieved by

x(t)

in this oscillation.