If a non linear system is in a state of periodic motion then we can find the period of the mtion using

(1)

where the last integral is expressed as a function of q using the Hamiltionian.

If the Hamiltonian has the formthen we can write so that

Substituting this into (1) gives

It must be noted here that the period is independent of the initial phase so the limits need only differ by

Example: The Hamiltonian of a vertical pendulum is given by

Ifthe pendulum has enough energy to describe a vertical circle and

after using the half angle formulae and putting