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 form
then 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
If
the pendulum has enough energy to describe a vertical circle and
after using the half angle formulae and putting