A particle of mass m moves in one dimension subject to the potential Obtain an integral ex...

A particle of mass m moves in one dimension subject to the potential

Obtain an integral expression for Hamilton’s characteristic function. Under what conditions can action-angle variables be used? Assuming these are met, find the frequency of oscillation by the action-angle method. (The integral for J can be evaluated by manipulating the integrand so that the square root appears in the denominator.) Check your result in the limit of oscillations of small amplitude.