Towards an invariant for the time-dependent anharmonic oscillator

The Hamiltonians H= (1/2)(p²+q²)+λ (t) q³ and $\mathrm{H̄}$ = (1/2)(P²+Q²)+Q³, where $\mathrm{\lambda ̄}$ (t) =1, are related by time‐dependent polynomial canonical transformations. Formulas are constructed for the generating function F₂(q,P,t) as well as for the direct relations between (Q,P) and (q,p). These formulas are expressed fairly concisely in terms of time integrals. The form is seen to be applicable to all polynomial transformations of time‐dependent anharmonic oscillator systems.