Classical Mechanics (Kibble & Berkshire) · Problem 8.7

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

Physics — Mechanics · phase-paths-damped-oscillator

Problem

A damped linear oscillator satisfies the equation x¨+x˙+x=0.\ddot{x} + \dot{x} + x = 0. Show that the polar equations for the motion of the phase points are r˙=−rsin⁡2θ,θ˙=−(1+12sin⁡2θ).\dot{r} = -r\sin^2\theta, \qquad \dot{\theta} = -\left(1 + \tfrac{1}{2}\sin 2\theta\right). Show that every phase path encircles the origin infinitely many times in the clockwise direction. Show further that these phase paths terminate at the origin.

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