The Adiabatic Invariant of the Linear or Nonlinear Oscillator

H. R. Lewis's treatment of the time-dependent harmonic oscillator is extended to the general time-dependent nonlinear oscillator. A prescription is given for obtaining an adiabatic invariant as a power series in the amplitude and to any order in the time variation of the coefficients. An exact invariant of the same algebraic form as the adiabatic invariant is also derived. The exact invariant can be so chosen that, at any particular time, it is equal to the adiabatic invariant. Comparison of the adiabatic and exact invariants permits a calculation of the nonadiabatic changes in the adiabatic invariant in the course of time. Applications to a linear and a nonlinear example are presented and the statistical distribution of the long time changes in the adiabatic invariants are determined in the two cases.