A certain oscillator with generalised coordinate q has Lagrangian Verify that q ∗ = sin 2t is…

A certain oscillator with generalised coordinate q has Lagrangian

Verify that q∗ = sin 2t is a motion of the oscillator, and show directly that it makes the action functional S[q ] satationary in any time interval [0, τ ]. For the time interval 0 ≤ t ≤ π, find the variation in the action functional corresponding to the variations (i) h = ϵ sin 4t, (ii) h = ϵ sin t, where ϵ is a small parameter. Deduce that the motion q∗ = sin 2t does not make S a minimum or a maximum.