Evolutionary force billiards

A. T. Fomenko ; V. V. Vedyushkina

Izvestiya. Mathematics , Tome 86 (2022) no. 5, pp. 943-979.

Résumé

A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems with two degrees of freedom on the entire symplectic four-dimensional phase manifold at a time, rather than on only individual isoenergy 3-surfaces. For instance, this occurs in the Euler and Lagrange cases. It has also been proved that these two well-known systems are “billiard-equivalent”, despite the fact that the former one is square integrable, and the latter one allows a linear integral.

Détail

Keywords: integrable system, evolutionary force billiards.
Mots-clés : billiard book, Fomenko–Zieschang invariant, Liouville equivalence

Bibliographie

[1] S. Smale, “Topology and mechanics. I”, Invent. Math., 10:4 (1970), 305–331 | DOI | MR | Zbl

[2] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Dokl. Akad. Nauk SSSR, 287:5 (1986), 1071–1075 ; English transl. Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[3] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Izv. Akad. Nauk SSSR Ser. Mat., 50:6 (1986), 1276–1307 ; English transl. Math. USSR-Izv., 29:3 (1987), 629–658 | MR | Zbl | DOI

[4] A. T. Fomenko and H. Zieschang, “On the topology of the three-dimensional manifolds arising in Hamiltonian mechanics”, Dokl. Akad. Nauk SSSR, 294:2 (1987), 283–287 ; English transl. Soviet Math. Dokl., 35:2 (1987), 529–534 | MR | Zbl

[5] A. T. Fomenko and H. Zieschang, “On typical topological properties of integrable Hamiltonian systems”, Izv. Akad. Nauk SSSR Ser. Mat., 52:2 (1988), 378–407 ; English transl. Math. USSR-Izv., 32:2 (1989), 385–412 | MR | Zbl | DOI

[6] 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. Math. USSR-Izv., 36:3 (1991), 567–596 | MR | Zbl | DOI

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

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

[9] V. V. Fokicheva and A. T. Fomenko, “Integrable billiards model important integrable cases of rigid body dynamics”, Dokl. Akad. Nauk, 465:2 (2015), 150–153 ; English transl. Dokl. Math., 92:3 (2015), 682–684 | DOI | MR | Zbl | DOI

[10] V. V. Fokicheva and A. T. Fomenko, “Billiard systems as the models for the rigid body dynamics”, Advances in dynamical systems and control, Stud. Syst. Decis. Control, 69, Springer, Cham, 2016, 13–33 | DOI | MR | Zbl

[11] 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. Izv. Math., 81:4 (2017), 688–733 | DOI | MR | Zbl | DOI

[12] V. V. Vedyushkina and A. T. Fomenko, “Reducing the degree of integrals of Hamiltonian systems by using billiards”, Dokl. Akad. Nauk, 486:2 (2019), 151–155 ; English transl. Dokl. Math., 99:3 (2019), 266–269 | DOI | Zbl | DOI

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

[14] V. V. Vedyushkina, “The Liouville foliation of nonconvex topological billiards”, Dokl. Akad. Nauk, 478:1 (2018), 7–11 ; English transl. Dokl. Math., 97:1 (2018), 1–5 | DOI | MR | Zbl | DOI

[15] 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. Akad. Nauk, 479:6 (2018), 607–610 ; English transl. Dokl. Math., 97:2 (2018), 174–176 | DOI | MR | Zbl | DOI

[16] V. V. Vedyushkina and I. S. Kharcheva, “Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems”, Mat. Sb., 209:12 (2018), 17–56 ; English transl. Sb. Math., 209:12 (2018), 1690–1727 | DOI | MR | Zbl | DOI

[17] A. A. Oshemkov, “Fomenko invariants for the main integrable cases of the rigid body motion equations”, Topological classification of integrable systems, Adv. Soviet Math., 6, Amer. Math. Soc., Providence, RI, 1991, 67–146 | DOI | MR | Zbl

[18] A. V. Bolsinov, P. H. Richter, and A. T. Fomenko, “The method of loop molecules and the topology of the Kovalevskaya top”, Mat. Sb., 191:2 (2000), 3–42 ; English transl. Sb. Math., 191:2 (2000), 151–188 | DOI | MR | Zbl | DOI

[19] P. V. Morozov, “The Liouville classification of integrable systems of the Clebsch case”, Mat. Sb., 193:10 (2002), 113–138 ; English transl. Sb. Math., 193:10 (2002), 1507–1533 | DOI | MR | Zbl | DOI

[20] P. V. Morozov, “Topology of Liouville foliations in the Steklov and the Sokolov integrable cases of Kirchhoff's equations”, Mat. Sb., 195:3 (2004), 69–114 ; English transl. Sb. Math., 195:3 (2004), 369–412 | DOI | MR | Zbl | DOI

[21] P. V. Morozov, “Calculation of the Fomenko–Zieschang invariants in the Kovalevskaya–Yehia integrable case”, Mat. Sb., 198:8 (2007), 59–82 ; English transl. Sb. Math., 198:8 (2007), 1119–1143 | DOI | MR | Zbl | DOI

[22] N. S. Slavina, “Topological classification of systems of Kovalevskaya–Yehia type”, Mat. Sb., 205:1 (2014), 105–160 ; English transl. Sb. Math., 205:1 (2014), 101–155 | DOI | MR | Zbl | DOI

[23] V. A. Kibkalo, “The topology of the analog of Kovalevskaya integrability case on the Lie algebra $so(4)$ under zero area integral”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2016, no. 3, 46–50 ; English transl. Moscow Univ. Math. Bull., 71:3 (2016), 119–123 | Zbl | DOI | MR

[24] V. Kibkalo, “Topological analysis of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra $\operatorname{so}(4)$”, Lobachevskii J. Math., 39:9 (2018), 1396–1399 | DOI | MR | Zbl

[25] V. A. Kibkalo, “Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra $\operatorname{so}(4)$”, Mat. Sb., 210:5 (2019), 3–40 ; English transl. Sb. Math., 210:5 (2019), 625–662 | DOI | MR | Zbl | DOI

[26] V. Kibkalo, “Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra $\operatorname{so}(3, 1)$”, Topology Appl., 275 (2020), 107028 | DOI | MR | Zbl

[27] D. A. Fedoseev and A. T. Fomenko, “Noncompact bifurcations of integrable dynamic systems”, Fundam. Prikl. Mat., 21:6 (2016), 217–243 ; English transl. J. Math. Sci. (N.Y.), 248:6 (2020), 810–827 | MR | Zbl | DOI

[28] S. S. Nikolaenko, “Topological classification of the Goryachev integrable systems in the rigid body dynamics: non-compact case”, Lobachevskii J. Math., 38:6 (2017), 1050–1060 | DOI | MR | Zbl

[29] V. A. Kibkalo, “Noncompactness property of fibers and singularities of non-Euclidean Kovalevskaya system on pencil of Lie algebras”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 6 (2020), 56–59 ; English transl. Moscow Univ. Math. Bull., 75:6 (2020), 263–267 | MR | Zbl | DOI

[30] E. A. Kudryavtseva, “An analogue of the Liouville theorem for integrable Hamiltonian systems with incomplete flows”, Dokl. Akad. Nauk, 445:4 (2012), 383–385 ; English transl. Dokl. Math., 86:1 (2012), 527–529 | MR | Zbl | DOI

[31] Nguyen Tien Zung, L. S. Polyakova, and E. N. Selivanova, “Topological classification of integrable geodesic flows on orientable two-dimensional Riemannian manifolds with additional integral depending on momenta linearly or quadratically”, Funktsional. Anal. i Prilozhen., 27:3 (1993), 42–56 ; English transl. Funct. Anal. Appl., 27:3 (1993), 186–196 | MR | Zbl | DOI

[32] E. N. Selivanova, “Classification of geodesic flows of Liouville metrics on the two-dimensional torus up to topological equivalence”, Mat. Sb., 183:4 (1992), 69–86 ; English transl. Sb. Math., 75:2 (1993), 491–505 | MR | Zbl | DOI

[33] V. V. Kalashnikov, “Topological classification of quadratic-integrable geodesic flows on a two-dimensional torus”, Uspekhi Mat. Nauk, 50:1(301) (1995), 201–202 ; English transl. Russian Math. Surveys, 50:1 (1995), 200–201 | MR | Zbl | DOI

[34] 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. Izv. Math., 83:6 (2019), 1137–1173 | DOI | MR | Zbl | DOI

[35] 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. Sb. Math., 207:3 (2016), 358–399 | DOI | MR | Zbl | DOI

[36] D. S. Timonina, “Liouville classification of integrable geodesic flows on a torus of revolution in a potential field”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2017, no. 3, 35–43 ; English transl. Moscow Univ. Math. Bull., 72:3 (2017), 121–128 | MR | Zbl | DOI

[37] 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 | MR | Zbl

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

[39] V. V. Vedyushkina and I. S. Kharcheva, “Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems”, Mat. Sb., 212:8 (2021), 89–150 ; English transl. Sb. Math., 212:8 (2021), 1122–1179 | DOI | MR | Zbl | DOI

[40] V. V. Vedyushkina, V. A. Kibkalo, and A. T. Fomenko, “Topological modeling of integrable systems by billiards: realization of numerical invariants”, Dokl. Akad. Nauk. Matem., inform., proc. upr., 493 (2020), 9–12 ; English transl. Dokl. Math., 102:1 (2020), 269–271 | DOI | Zbl | DOI

[41] V. V. Vedyushkina and V. A. Kibkalo, “Realization of the numerical invariant of the Seifert fibration of integrable systems by billiards”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2020, no. 4, 22–28 ; English transl. Moscow Univ. Math. Bull., 75:4 (2020), 161–168 | MR | Zbl | DOI

[42] V. V. Vedyushkina, “Integrable billiard systems realize toric foliations on lens spaces and the 3-torus”, Mat. Sb., 211:2 (2020), 46–73 ; English transl. Sb. Math., 211:2 (2020), 201–225 | DOI | MR | Zbl | DOI

[43] F. Waldhausen, “Eine Klasse von 3-dimensionalen Mannigfaltighkeiten. I”, Invent. Math., 3:4 (1967), 308–333 | DOI | MR | Zbl

[44] F. Waldhausen, “Eine Klasse von 3-dimensionalen Mannigfaltighkeiten. II”, Invent. Math., 4:2 (1967), 88–117 | DOI | MR | Zbl

[45] V. V. Vedyushkina, “The Liouville foliation of the billiard book modelling the Goryachev–Chaplygin case”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2020, no. 1, 64–68 ; English transl. Moscow Univ. Math. Bull., 75:1 (2020), 42–46 | MR | Zbl | DOI

[46] 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

[47] E. E. Karginova, “Billiards bounded by arcs of confocal quadrics on the Minkowski plane”, Mat. Sb., 211:1 (2020), 3–31 ; English transl. Sb. Math., 211:1 (2020), 1–28 | DOI | MR | Zbl | DOI

[48] I. F. Kobtsev, “An elliptic billiard in a potential force field: classification of motions, topological analysis”, Mat. Sb., 211:7 (2020), 93–120 ; English transl. Sb. Math., 211:7 (2020), 987–1013 | DOI | MR | Zbl | DOI

[49] S. E. Pustovoytov, “Topological analysis of a billiard in elliptic ring in a potential field”, Fundam. Prikl. Mat., 22:6 (2019), 201–225 ; English transl. J. Math. Sci. (N.Y.), 259:5 (2021), 712–729 | MR | Zbl | DOI

[50] S. E. Pustovoitov, “Topological analysis of a billiard bounded by confocal quadrics in a potential field”, Mat. Sb., 212:2 (2021), 81–105 ; English transl. Sb. Math., 212:2 (2021), 211–233 | DOI | MR | Zbl | DOI

[51] G. V. Belozerov, “Topological classification of integrable geodesic billiards on quadrics in three-dimensional Euclidean space”, Mat. Sb., 211:11 (2020), 3–40 ; English transl. Sb. Math., 211:11 (2020), 1503–1538 | DOI | MR | Zbl | DOI

[52] I. F. Kobtsev, “The geodesic flow on a two-dimensional ellipsoid in the field of an elastic force. Topological classification of solutions”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2018, no. 2, 27–33 ; English transl. Moscow Univ. Math. Bull., 73:2 (2018), 64–70 | MR | Zbl | DOI

[53] V. V. Kozlov and D. V. Treshchev, Billiards. A genetic introduction to the dynamics of systems with impacts, Moscow Univ. press, Moscow, 1991 ; English transl. Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991 | MR | Zbl | DOI | MR | Zbl

[54] S. Tabachnikov, Geometry and billiards, Stud. Math. Libr., 30, Amer. Math. Soc., Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005 | DOI | MR | Zbl

[55] V. Dragović and M. Radnović, Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics, "Research Centre “Regular and Chaotic Dynamics”, Moscow–Izhevsk, 2010; English transl. Front. Math., Birkhäuser/Springer Basel AG, Basel, 2011 | DOI | MR | Zbl

[56] A. A. Glutsyuk, “On two-dimensional polynomially integrable billiards on surfaces of constant curvature”, Dokl. Akad. Nauk, 481:6 (2018), 594–598 ; English transl. Dokl. Math., 98:1 (2018), 382–385 | DOI | Zbl | DOI

[57] M. Bialy and A. E. Mironov, “Angular billiard and algebraic Birkhoff conjecture”, Adv. Math., 313 (2017), 102–126 | DOI | MR | Zbl

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

[59] M. Bialy and A. E. Mironov, “Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane”, Uspekhi Mat. Nauk, 74:2(446) (2019), 3–26 ; English transl. Russian Math. Surveys, 74:2 (2019), 187–209 | DOI | MR | Zbl | DOI

[60] A. Avila, J. De Simoi, and V. Kaloshin, “An integrable deformation of an ellipse of small eccentricity is an ellipse”, Ann. of Math. (2), 184:2 (2016), 527–558 | DOI | MR | Zbl

[61] V. Kaloshin and A. Sorrentino, “On the local Birkhoff conjecture for convex billiards”, Ann. of Math. (2), 188:1 (2018), 315–380 | DOI | MR | Zbl

[62] Nguyen Tien Zung, “Symplectic topology of integrable Hamiltonian systems. II. Topological classification”, Compositio Math., 138:2 (2003), 125–156 | DOI | MR | Zbl

[63] Nguyen Tien Zung, “Symplectic topology of integrable Hamiltonian systems. I. Arnold–Liouville with singularities”, Compositio Math., 101:2 (1996), 179–215 | MR | Zbl

[64] A. Bolsinov, L. Guglielmi, and E. Kudryavtseva, “Symplectic invariants for parabolic orbits and cusp singularities of integrable systems”, Philos. Trans. Roy. Soc. A, 376:2131 (2018), 20170424 | DOI | MR | Zbl

[65] E. A. Kudryavtseva and N. N. Martynchuk, “Existence of a smooth Hamiltonian circle action near parabolic orbits and cuspidal tori”, Regul. Chaotic Dyn., 26:6 (2021), 732–741 | DOI | MR | Zbl

[66] V. V. Kalashnikov, “Typical integrable Hamiltonian systems on a four-dimensional symplectic manifold”, Izv. Ross. Akad. Nauk Ser. Mat., 62:2 (1998), 49–74 ; English transl. Izv. Math., 62:2 (1998), 261–285 | DOI | MR | Zbl | DOI

[67] E. A. Kudryavtseva, “Hidden toric symmetry and structural stability of singularities in integrable systems”, Eur. J. Math., 2021, 1–63, Publ. online | DOI

[68] A. V. Bolsinov and A. A. Oshemkov, “Singularities of integrable Hamiltonian systems”, Topological methods in the theory of integrable systems, Camb. Sci. Publ., Cambridge, 2006, 1–67 | MR | Zbl

[69] A. A. Oshemkov and M. A. Tuzhilin, “Integrable perturbations of saddle singularities of rank 0 of integrable Hamiltonian systems”, Mat. Sb., 209:9 (2018), 102–127 ; English transl. Sb. Math., 209:9 (2018), 1351–1375 | DOI | MR | Zbl | DOI

[70] A. Bolsinov and A. Izosimov, “Smooth invariants of focus-focus singularities and obstructions to product decomposition”, J. Symplectic Geom., 17:6 (2019), 1613–1648 | DOI | MR | Zbl

[71] I. K. Kozlov and A. A. Oshemkov, “Classification of saddle-focus singularities”, Chebyshevskii Sb., 21:2 (2020), 228–243 | MR | Zbl

[72] S. S. Nikolaenko, “Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds”, Mat. Sb., 211:8 (2020), 68–101 ; English transl. Sb. Math., 211:8 (2020), 1127–1158 | DOI | MR | Zbl | DOI

[73] A. T. Fomenko and V. A. Kibkalo, “Saddle singularities in integrable Hamiltonian systems: examples and algorithms”, Contemporary approaches and methods in fundamental mathematics and mechanics, Underst. Complex Syst., Springer, Cham, 2021, 3–26 | DOI | MR | Zbl

[74] I. S. Kharcheva, “Isoenergetic manifolds of integrable billiard books”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2020, no. 4, 12–22 ; English transl. Moscow Univ. Math. Bull., 75:4 (2020), 149–160 | MR | Zbl | DOI