Home > Proceedings > Geom. Integrability & Quantization > Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization > Bertrand Systems on Spaces of Constant Sectional Curvature. The Action-Angle Analysis. Classical, Quasi-Classical and Quantum Problems
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VOL. 16 | 2015 Bertrand Systems on Spaces of Constant Sectional Curvature. The Action-Angle Analysis. Classical, Quasi-Classical and Quantum Problems
Jan J. Sławianowski, Barbara Gołubowska

Editor(s) Ivaïlo M. Mladenov, Andrei Ludu, Akira Yoshioka

## Abstract

Studied is the problem of degeneracy of mechanical systems the configuration space of which is the three-dimensional sphere, the elliptic space, i.e., the quotient of that sphere modulo the antipodal identification, and finally, the three-dimensional pseudo-sphere, namely, the Lobatchevski space. In other words, discussed are systems on groups ${\rm{SU}}(2)$, ${{\rm{SO}}}(3,\mathbb{R})$, and ${\rm{SL}}(2,\mathbb{R})$ or its quotient ${{\rm{SO}}}(1,2)$. The main subject are completely degenerate Bertrand-like systems. We present the action-angle classical description, the corresponding quasi-classical analysis and the rigorous quantum formulas. It is interesting that both the classical action-angle formulas and the rigorous quantum mechanical energy levels are superpositions of the flat-space expression, with those describing free geodetic motion on groups.

## Information

Published: 1 January 2015
First available in Project Euclid: 13 July 2015

zbMATH: 1352.53009
MathSciNet: MR3363840

Digital Object Identifier: 10.7546/giq-16-2015-110-138