Geodesic
Research of the structure of the Liouville foliation of an integrable elliptical billiard with polynomial potential
Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 62-102.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider a planar billiard bounded by an ellipse in the potential force field. An explicit formula of the polynomial potential preserving integrability of such a billiard was found. The structure of the Liouville foliation at all non singular energy levels was studied using the method of separation of variables. Namely, an algorithm that constructs the bifurcation diagram and the Fomenko-Zieschang invariants from the values of the parameters of the potential was proposed. In addition, the topology of the isoenergetic manifold was studied and the cases of rigid body dynamics, which are Liouville equivalent to our billiard, were established.
Keywords: integrable Hamiltonian system, billiard, polynomial potential, Liouville foliation, Fomenko-Zieschang invariant. 
@article{CHEB_2024_25_1_a5,
     author = {S. E. Pustovoitov},
     title = {Research of the structure of the {Liouville} foliation of an integrable elliptical billiard with polynomial potential},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {62--102},
     publisher = {mathdoc},
     volume = {25},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2024_25_1_a5/}
}
TY  - JOUR
AU  - S. E. Pustovoitov
TI  - Research of the structure of the Liouville foliation of an integrable elliptical billiard with polynomial potential
JO  - Čebyševskij sbornik
PY  - 2024
SP  - 62
EP  - 102
VL  - 25
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2024_25_1_a5/
LA  - ru
ID  - CHEB_2024_25_1_a5
ER  - 
%0 Journal Article
%A S. E. Pustovoitov
%T Research of the structure of the Liouville foliation of an integrable elliptical billiard with polynomial potential
%J Čebyševskij sbornik
%D 2024
%P 62-102
%V 25
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2024_25_1_a5/
%G ru
%F CHEB_2024_25_1_a5
S. E. Pustovoitov. Research of the structure of the Liouville foliation of an integrable elliptical billiard with polynomial potential. Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 62-102. http://geodesic.mathdoc.fr/item/CHEB_2024_25_1_a5/

[1] Birkhoff, G., Dynamical systems, American Mathematical Society Colloquium publications, IX, 1966, 305 pp. | MR

[2] Clebsch, A. (ed.), Jacobi's lectures on dynamics, 2nd edition, Hindustan book agency, New Delhi, 2009, 339 pp. | MR

[3] V. Kaloshin, A. Sorrentino, “On the local Birkhoff conjecture for convex billiards”, Ann. of Math., 188:1 (2018), 315–380 | DOI | MR | Zbl

[4] A. A. Glutsyuk, “On polynomially integrable Birkhoff billiards on surfaces of constant curvature”, Journal of the European Mathematical Society, 23:3 (2021), 995–1049 | DOI | MR | Zbl

[5] Fokicheva, V. V., “A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics”, Sb. Math., 206:10 (2015), 1463–1507 | DOI | DOI | MR | Zbl

[6] V. V. Vedyushkina, I. S. Kharcheva, “Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems”, Sb. Math., 209:12 (2018), 1690–1727 | DOI | DOI | MR | Zbl

[7] Kibkalo V. A., Fomenko A. T., Kharcheva I. S., “Realization of integrable Hamiltonial systems by billiard books”, Transactions of the Moscow Mathematical Society, 80, 2021 | MR | Zbl

[8] M. Bialy, A. E. Mironov, “Algebraic non-integrability of magnetic billiards”, J. Phys. A, 49:45 (2016), 455101, 18 pp. | DOI | MR | Zbl

[9] Vedyushkina V. V., Pustovoitov S. E., “Classification of liouville foliations of integrable topological billiards in magnetic fields”, Sbornik Mathematics, 214:2 (2023), 166–196 | DOI | DOI | MR | Zbl

[10] J. Appl. Math. Mech., 59:1 (1995), 1–7 | DOI | MR | Zbl

[11] V. I. Dragovich, “Integrable perturbations of a Birkhoff billiards inside an ellipse”, J. Appl. Maths Mechs., 62:1 (1998), 159–162 | DOI | MR | Zbl

[12] Kobtsev, I. F., “An elliptic billiard in a potential force field: classification of motions, topological analysis”, Sb. Math., 211:7 (2020), 987–1013 | DOI | DOI | MR | Zbl

[13] Pustovoitov, S. E., “Topological Analysis of an Elliptic Billiard in a Fourth-Order Potential Field”, Moscow University Mathematics Bulletin, 76:5 (2021), 193–205 | DOI | MR | Zbl

[14] Bolsinov, A. V., Fomenko, A. T., Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, 2004 | MR | Zbl

[15] Kozlov, V. V., Treshchev, D. V., Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts, Translations of Mathematical Monographs, 89, American Mathematical Society, 1991, 171 pp. | DOI | MR | Zbl

[16] Kharlamov M.P., “Topological analysis and Boolean function: I. Methods and applications to the classical systems”, Non-lineal dynamics, 6:4 (2010), 769–805

[17] Fomenko, A. T., “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions of integrability”, Mathematics of the USSR-Izvestiya, 29:3 (1987), 629–658 | DOI | MR | Zbl | Zbl

[18] Fomenko, A. T., “The symplectic topology of completely integrable Hamiltonian systems”, Russian Mathematical Surveys, 44:1 (1989), 181 | DOI | MR | Zbl

[19] Fomenko, A. T., Zieschang, H., “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Math. USSR-Izv., 36:3 (1990), 567–596 | DOI | MR | Zbl

[20] Fomenko, A. T., “A bordism theory for integrable nondegenerate Hamiltonian systems with two degrees of freedom. A new topological invariant of higherdimensional integrable systems”, Math. USSR-Izv., 39:1 (1992), 731–759 | DOI | MR