Canonical perturbation expansions to large order from classical hypervirial and Hellmann–Feynman theorems

The classical hypervirial and Hellmann–Feynman theorems are used to formulate a ‘‘perturbation theory without Fourier series’’ that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical Hamiltonians. As in the case where these theorems are used to generate quantum mechanical Rayleigh–Schrödinger perturbation series, the method is very efficient and may be used to generate expansions to large order either numerically or in algebraic form. Here, the method is applied to one-dimensional anharmonic oscillators and radial Kepler problems. In all cases, the classical series for energies and expectation values are seen to correspond to the expansions associated with their quantum mechanical counterparts through an appropriate action preserving classical limit as discussed by Turchetti, Graffi, and Paul. This ‘‘action fixing’’ is inherent in the classical Hellmann–Feynman theorem applied to periodic orbits.