Proceedings of the 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

Jan J. Sławianowski and Barbara Gołubowska

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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.

Article information

Source
Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, Andrei Ludu and Akira Yoshioka, eds. (Sofia: Avangard Prima, 2015), 110-138

Dates
First available in Project Euclid: 13 July 2015

Permanent link to this document
https://projecteuclid.org/ euclid.pgiq/1436815739

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

Mathematical Reviews number (MathSciNet)
MR3363840

Zentralblatt MATH identifier
1352.53009

Citation

Sławianowski, Jan J.; Gołubowska, Barbara. Bertrand Systems on Spaces of Constant Sectional Curvature. The Action-Angle Analysis. Classical, Quasi-Classical and Quantum Problems. Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization, 110--138, Avangard Prima, Sofia, Bulgaria, 2015. doi:10.7546/giq-16-2015-110-138. https://projecteuclid.org/euclid.pgiq/1436815739


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