Geodesic
Classification of Liouville foliations of integrable topological billiards in magnetic fields
Sbornik. Mathematics, Tome 214 (2023) no. 2, pp. 166-196 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The topology of the Liouville foliations of integrable magnetic topological billiards, systems in which a ball moves on piecewise smooth two-dimensional surfaces in a constant magnetic field, is considered. The Fomenko-Zieschang invariants of Liouville equivalence are calculated for the Hamiltonian systems arising, and the topology of invariant 3-manifolds, isointegral and isoenergy ones, is investigated. The Liouville equivalence of such billiards to some known Hamiltonian systems is discovered, for instance, to the geodesic flows on 2-surfaces and to systems of rigid body dynamics. In particular, peculiar saddle singularities are discovered in which singular circles have different orientations — such systems were also previously encountered in mechanical systems in a magnetic field on surfaces of revolution homeomorphic to a 2-sphere. Bibliography: 13 titles.
Keywords: integrable systems, magnetic field, topological billiard
Mots-clés : Liouville foliation, Fomenko-Zieschang invariant. 
@article{SM_2023_214_2_a1,
     author = {V. V. Vedyushkina and S. E. Pustovoitov},
     title = {Classification of {Liouville} foliations of integrable topological billiards in magnetic fields},
     journal = {Sbornik. Mathematics},
     pages = {166--196},
     year = {2023},
     volume = {214},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2023_214_2_a1/}
}
TY  - JOUR
AU  - V. V. Vedyushkina
AU  - S. E. Pustovoitov
TI  - Classification of Liouville foliations of integrable topological billiards in magnetic fields
JO  - Sbornik. Mathematics
PY  - 2023
SP  - 166
EP  - 196
VL  - 214
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_2023_214_2_a1/
LA  - en
ID  - SM_2023_214_2_a1
ER  - 
%0 Journal Article
%A V. V. Vedyushkina
%A S. E. Pustovoitov
%T Classification of Liouville foliations of integrable topological billiards in magnetic fields
%J Sbornik. Mathematics
%D 2023
%P 166-196
%V 214
%N 2
%U http://geodesic.mathdoc.fr/item/SM_2023_214_2_a1/
%G en
%F SM_2023_214_2_a1
V. V. Vedyushkina; S. E. Pustovoitov. Classification of Liouville foliations of integrable topological billiards in magnetic fields. Sbornik. Mathematics, Tome 214 (2023) no. 2, pp. 166-196. http://geodesic.mathdoc.fr/item/SM_2023_214_2_a1/

[1] V. V. Fokicheva (Vedyushkina), “A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics”, Mat. Sb., 206:10 (2015), 127–176 ; English transl. in Sb. Math., 206:10 (2015), 1463–1507 | DOI | MR | Zbl | DOI

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

[3] V. V. Vedyushkina, “The Fomenko-Zieschang invariants of nonconvex topological billiards”, Mat. Sb., 210:3 (2019), 17–74 ; English transl. in Sb. Math., 210:3 (2019), 310–363 | DOI | MR | Zbl | DOI

[4] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, v. 1, 2, Udmurtian University Publishing House, Izhevsk, 1999, 444 pp., 447 pp. ; English transl., Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | MR | Zbl | DOI | MR | Zbl

[5] E. A. Kudryavtseva and A. A. Oshemkov, “Bifurcations of integrable mechanical systems with magnetic field on surfaces of revolution”, Chebyshevskii Sb., 21:2 (2020), 244–265 (Russian) | DOI | MR | Zbl

[6] A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Uspekhi Mat. Nauk, 44:1(265) (1989), 145–173 ; English transl. in Russian Math. Surveys, 44:1 (1989), 181–219 | MR | Zbl | DOI

[7] A. T. Fomenko and H. Zieschang, “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Izv. Akad. Nauk SSSR Ser. Mat., 54:3 (1990), 546–575 ; English transl. in Math. USSR-Izv., 36:3 (1991), 567–596 | MR | Zbl | DOI

[8] A. T. Fomenko and V. V. Vedyushkina, “Billiards and integrability in geometry and physics. New scope and new potential”, Vestn. Moskov. Univ. Ser. 1 Mat. Mekh., 2019, no. 3, 15–25 ; English transl. in Moscow Univ. Math. Bull., 74:3 (2019), 98–107 | MR | Zbl | DOI

[9] V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Ross. Akad. Nauk Ser. Mat., 83:6 (2019), 63–103 ; English transl. in Izv. Math., 83:6 (2019), 1137–1173 | DOI | MR | Zbl | DOI

[10] A. T. Fomenko and V. V. Vedyushkina, “Implementation of integrable systems by topological, geodesic billiards with potential and magnetic field”, Russ. J. Math. Phys., 26:3 (2019), 320–333 | DOI | MR | Zbl

[11] V. V. Vedyushkina, V. A. Kibkalo and A. T. Fomenko, “Topological modeling of integrable systems by billiards: realization of numerical invariants”, Dokl. Ross. Akad. Nauk Mat. Inform. Protsessy Upravl., 493 (2020), 9–12 ; English transl. in Dokl. Math., 102:1 (2020), 269–271 | DOI | Zbl | DOI | MR

[12] V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Ross. Akad. Nauk Ser. Mat., 81:4 (2017), 20–67 ; English transl. in Izv. Math., 81:4 (2017), 688–733 | DOI | MR | Zbl | DOI

[13] E. O. Kantonistova, “Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution”, Mat. Sb., 207:3 (2016), 47–92 ; English transl. in Sb. Math., 207:3 (2016), 358–399 | DOI | MR | Zbl | DOI