Classical Mechanics (Kibble & Berkshire) · Problem 13.11

Hamilton's equations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · hamiltons-principle

Problem

A certain oscillator with generalised coordinate qq has Lagrangian L=q˙2−4q2.L = \dot{q}^2 - 4q^2. Verify that q∗=sin⁡2tq^* = \sin 2t is a motion of the oscillator, and show directly that it makes the action functional S[q]S[q] stationary in any time interval [0,τ][0, \tau]. For the time interval 0≤t≤π0 \leq t \leq \pi, find the variation in the action functional corresponding to the variations (i) h=ϵsin⁡4th = \epsilon \sin 4t, (ii) h=ϵsin⁡th = \epsilon \sin t, where ϵ\epsilon is a small parameter. Deduce that the motion q∗=sin⁡2tq^* = \sin 2t does not make SS a minimum or a maximum.
Intuitive explanation

The action functional is stationary when the Euler-Lagrange equation is satisfied—this is the principle of least action. To verify q∗q^* is a true motion, substitute it into the equation of motion and check it works. Then, to show stationarity directly, compute the first variation δS\delta S by perturbing q∗q^* slightly and confirming the linear term in ϵ\epsilon vanishes. Finally, test two different perturbations: if one increases the action and another decreases it, the stationary point is a saddle, not an extremum—revealing the deep difference between "stationary" (zero first variation) and "extremum" (definite sign of second variation).

Intuitive explanation

The action functional is stationary when the Euler-Lagrange equation is satisfied—this is the principle of least action. To verify q∗q^* is a true motion, substitute it into the equation of motion and check it works. Then, to show stationarity directly, compute the first variation δS\delta S by perturbing q∗q^* slightly and confirming the linear term in ϵ\epsilon vanishes. Finally, test two different perturbations: if one increases the action and another decreases it, the stationary point is a saddle, not an extremum—revealing the deep difference between "stationary" (zero first variation) and "extremum" (definite sign of second variation).

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