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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

Rights: Copyright © 2015 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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