Classical Mechanics (Kibble & Berkshire) · Problem 14.11

The general theory of small oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · liouville-damped-oscillator

Problem

A one dimensional damped oscillator with coordinate qq satisfies the equation q¨+4q˙+3q=0\ddot{q} + 4\dot{q} + 3q = 0, which is equivalent to the first order system q˙=v,v˙=−3q−4v.\dot{q} = v, \qquad \dot{v} = -3q - 4v. Show that the area a(t)a(t) of any region of points moving in (q,v)(q, v)-space has the time variation a(t)=a(0) e−4t.a(t) = a(0)\,e^{-4t}. Does this result contradict Liouville's theorem?

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