Canonical perturbation expansions to large order from classical hypervirial and Hellmann-Feynman theorems. (English) Zbl 0765.70013
The classical hypervirial and Hellmann-Feynman theorems [R. P. Feynman, Phys. Rev., II. Ser. 56, 340-343 (1939; Zbl 0022.42302), and H. Hellmann, Einführung in die Quantenchemie (1937)] 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.
MSC:
| 70H99 | Hamiltonian and Lagrangian mechanics |
| 70F05 | Two-body problems |
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
Keywords:
energies of perturbed periodic orbits; separable classical Hamiltonians; quantum mechanical Rayleigh-Schrödinger perturbation series; one-dimensional anharmonic oscillators; radial Kepler problemsCitations:
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