  In this letter, we study the purely nonlinear oscillator by the method of
action-angle variables of Hamiltonian systems. The frequency of the
non-isochronous system is obtained, which agrees well with the previously known
result. Exact analytic solutions of the system involving generalized
trigonometric functions are presented. We also present arguments to show the
adiabatic invariance of the action variable for a time-dependent purely
nonlinear oscillator.
