Abstract
We consider a class of integrable Hamiltonian systems with two degrees of freedom, i.e., billiard books, which are a generalization of billiard in domains bounded by arcs of confocal quadrics. When studying billiard, first of all, the question arises about the topology of the phase space and the isoenergetic manifold. We prove that the phase space and the isoenergetic manifold in the case of billiard books are actually piecewise smooth manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamic of Systems with Impacts (Amer. Math. Soc., Providence, RI, 1991; Moscow State Univ., Moscow, 1991).
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification (CRC Press, Boca Raton, FL, 2004; NITs RKhD, Izhevsk, 1999). doi 10.1201/9780203643426
V. Dragović and M. Radnović, ‘‘Bifurcations of Liouville tori in elliptical billiards,’’ Regular Chaotic Dyn. 14, 479–494 (2009). doi 10.1134/S1560354709040054
V. Dragović and M. Radnović, Poncelet Porisms and Beyond (Springer, Basel, 2011). doi 10.1007/978-3-0348-0015-0
V. V. Fokicheva, ‘‘Description of singularities for system ‘billiard in an ellipse’,’’ Moscow Univ. Math. Bull. 67, 217–220 (2012). doi 10.3103/S0027132212050063
V. V. Fokicheva, ‘‘Classification of billiard motions in domains bounded by confocal parabolas,’’ Sb. Math. 205, 1201–1221 (2014). doi 10.1070/SM2014v205n08ABEH004415
V. V. Fokicheva, ‘‘Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas,’’ Moscow Univ. Math. Bull. 69, 148–158 (2014). doi 10.3103/S0027132214040020
V. V. Fokicheva and A. T. Fomenko, ‘‘Integrable billiards model important integrable cases of rigid body dynamics,’’ Dokl. Math. 92, 682–684 (2015). doi 10.1134/S1064562415060095
V. V. Vedyushkina and I. S. Kharcheva, ‘‘Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems,’’ Sb. Math. 209, 1690–1727 (2018). doi 10.1070/SM9039
V. V. Vedyushkina, A. T. Fomenko, and I. S. Kharcheva, ‘‘Modeling nondegenerate bifurcations of closures of solutions for integrable systems with two degrees of freedom by integrable topological billiards,’’ Dokl. Math. 97, 174–176 (2018). doi 10.1134/S1064562418020230
V. V. Fokicheva, ‘‘A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics,’’ Sb. Math. 206, 1463–1507 (2015). doi 10.1070/SM2015v206n10ABEH004502
V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, ‘‘Integrable topological billiards and equivalent dynamical systems,’’ Izv. Math. 81, 688–733 (2017). doi 10.1070/IM8602
ACKNOWLEDGMENTS
We thank supervisors A. T. Fomenko and V. V. Vedyushkina for the formulation of the problem, significant help, and constant attention to the work and to E. A. Kudryavtseva and A. A. Oshemkov for numerous useful comments.
Funding
The author is a scholar of the Theoretical Physics and Mathematics Advancement Foundation BASIS, agreement no. 18–2–6–50–1.
This work was supported by the program Leading Scientific SchoolsP of the Russian Federation (project no. NSH–6399.2018.1, agreement no. 075–02–2018–867).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by O. Pismenov
About this article
Cite this article
Kharcheva, I.S. Isoenergetic Manifolds of Integrable Billiard Books. Moscow Univ. Math. Bull. 75, 149–160 (2020). https://doi.org/10.3103/S0027132220040026
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132220040026