## 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

#### 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

Dates
First available in Project Euclid: 13 July 2015

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