Bertrand systems on spaces of constant sectional curvature. The action—angle analysis

Abstract

Discussed are Bertrand models on three-dimensional spaces of constant curvature. By “Bertrand” we mean isotropic potential models for which all bounded orbits are closed. The special stress is laid on the analysis in terms of action-angle variables and the corresponding problems of quasiclassical analysis and quantization. It is shown that from the action-angle point of view these models are described by formulae additively combining the corresponding expressions in Euclidean space and expressions characterizing free geodetic motion. Certain “cosmological” aspects of the problem are mentioned.