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 Izv. Akad. Nauk SSSR Ser. Mat., 1982, Volume 46, Issue 5, Pages 994–1010 (Mi izv1656)

Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities

V. N. Kolokoltsov

Abstract: In the paper an explicit description is given for all Riemannian metrics on the sphere and on the torus whose geodesic flows have an additional first integral that is both quadratic in the velocities and independent of the energy integral. Moreover, it is proved that on compact two-dimensional manifolds of higher genus the geodesic flows have no additional polynomial integral. All the results admit straightforward generalizations to arbitrary natural systems given on cotangent bundles of two-dimensional manifolds.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1983, 21:2, 291–306

Bibliographic databases:

UDC: 513.88
MSC: Primary 58F17, 53C22; Secondary 34C35, 58F07

Citation: V. N. Kolokoltsov, “Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities”, Izv. Akad. Nauk SSSR Ser. Mat., 46:5 (1982), 994–1010; Math. USSR-Izv., 21:2 (1983), 291–306

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Kozlov, “Integrability and non-integrability in Hamiltonian mechanics”, Russian Math. Surveys, 38:1 (1983), 1–76
2. V. V. Trofimov, A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras”, Russian Math. Surveys, 39:2 (1984), 1–67
3. A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658
4. I. A. Taimanov, “Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds”, Math. USSR-Izv., 30:2 (1988), 403–409
5. M. L. Byalyi, “First integrals that are polynomial in momenta for a mechanical system on a two-dimensional torus”, Funct. Anal. Appl., 21:4 (1987), 310–312
6. E. N. Selivanova, “Classification of geodesic flows of Liouville metrics on the two-dimensional torus up to topological equivalence”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 491–505
7. Nguyen Tien Zung, “The complexity of integrable Hamiltonian systems on a prescribed three-dimensional constant-energy submanifold”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 507–533
8. V. V. Kozlov, N. V. Denisova, “Symmetries and the topology of dynamical systems with two degrees of freedom”, Russian Acad. Sci. Sb. Math., 80:1 (1995), 105–124
9. Nguyen Tien Zung, L. S. Polyakova, E. N. Selivanova, “Topological Classification of Integrable Geodesic Flows on Orientable Two-Dimensional Riemannian Manifolds with Additional Integral Depending on Momenta Linearly or Quadratically”, Funct. Anal. Appl., 27:3 (1993), 186–196
10. V. V. Kozlov, N. V. Denisova, “Polynomial integrals of geodesic flows on a two-dimensional torus”, Russian Acad. Sci. Sb. Math., 83:2 (1995), 469–481
11. A. V. Bolsinov, A. T. Fomenko, “Integrable geodesic flows on the sphere, generated by Goryachev–Chaplygin and Kowalewski systems in the dynamics of a rigid body”, Math. Notes, 56:2 (1994), 859–861
12. K Rosquist, G Pucacco, J Phys A Math Gen, 28:11 (1995), 3235
13. I. K. Babenko, N. N. Nekhoroshev, “On complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral”, Math. Notes, 58:5 (1995), 1129–1135
14. V. V. Kalashnikov, “Topological classification of quadratic-integrable geodesic flows on a two-dimensional torus”, Russian Math. Surveys, 50:1 (1995), 200–201
15. A. V. Bolsinov, V. V. Kozlov, A. T. Fomenko, “The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body”, Russian Math. Surveys, 50:3 (1995), 473–501
16. V. V. Kozlov, V. V. Ten, “Topology of domains of possible motions of integrable systems”, Sb. Math., 187:5 (1996), 679–684
17. Ya. B. Vorobets, “Asymptotics of the spectrum of the Laplace–Beltrami operator on tori with Liouville and infra-Liouville metrics”, Russian Math. Surveys, 52:2 (1997), 430–431
18. N. V. Denisova, “The structure of infinitesimal symmetries of geodesic flows on a two-dimensional torus”, Sb. Math., 188:7 (1997), 1055–1069
19. A. V. Bolsinov, V. S. Matveev, A. T. Fomenko, “Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry”, Sb. Math., 189:10 (1998), 1441–1466
20. N. V. Denisova, “Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus”, Math. Notes, 64:1 (1998), 31–37
21. S. Yu. Dobrokhotov, A. I. Shafarevich, “Tunnel Splitting of the Spectrum of the Beltrami–Laplace Operators on Two-Dimensional Surfaces with Square Integrable Geodesic Flow”, Funct. Anal. Appl., 34:2 (2000), 133–134
22. N. V. Denisova, V. V. Kozlov, “Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space”, Sb. Math., 191:2 (2000), 189–208
23. A. V. Bolsinov, I. A. Taimanov, “Integrable Geodesic Flows on the Suspensions of Toric Automorphisms”, Proc. Steklov Inst. Math., 231 (2000), 42–58
24. A. V. Bolsinov, B. Jovanović, “Integrable geodesic flows on homogeneous spaces”, Sb. Math., 192:7 (2001), 951–968
25. Max Karlovini, Giuseppe Pucacco, Kjell Rosquist, Lars Samuelsson, “A unified treatment of quartic invariants at fixed and arbitrary energy”, J Math Phys (N Y ), 43:8 (2002), 4041
26. Vladimir S. Matveev, “Three-dimensional manifolds having metrics with the same geodesics”, Topology, 42:6 (2003), 1371
27. Bolsinov A.V., “Integrable geodesic flows on Riemannian manifolds: Construction and obstructions”, Proceedings of the Workshop on Contemporary Geometry and Related Topics, 2004, 57–103
28. Giuseppe Pucacco, Kjell Rosquist, “Configurational invariants of Hamiltonian systems”, J Math Phys (N Y ), 46:5 (2005), 052902
29. Alexey V. Bolsinov, Vladimir S. Matveev, Giuseppe Pucacco, “Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta”, Journal of Geometry and Physics, 59:7 (2009), 1048
30. GIUSEPPE PUCACCO, KJELL ROSQUIST, “NONSTANDARD SEPARABILITY ON THE Minkowski PLANE”, J. Nonlinear Math. Phys, 16:04 (2009), 421
31. Vladimir S. Matveev, Vsevolod V. Shevchishin, “Differential invariants for cubic integrals of geodesic flows on surfaces”, Journal of Geometry and Physics, 60:6-8 (2010), 833
32. V. T. Lisitsa, “On the conditions of total resonance of Liouville type Hamiltonian systems with $n$ degrees of freedom”, Zhurn. matem. fiz., anal., geom., 6:3 (2010), 295–304
33. Vladimir S. Matveev, Vsevolod V. Shevchishin, “Two-dimensional superintegrable metrics with one linear and one cubic integral”, Journal of Geometry and Physics, 61:8 (2011), 1353
34. Vladimir S. Matveev, “Two-dimensional metrics admitting precisely one projective vector field”, Math. Ann, 2011
35. Thomas J. Waters, “Regular and irregular geodesics on spherical harmonic surfaces”, Physica D: Nonlinear Phenomena, 2011
36. Misha Bialy, A.E.. Mironov, “Integrable geodesic flows on 2-torus: Formal solutions and variational principle”, Journal of Geometry and Physics, 2014
37. Giuseppe Pucacco, “Polynomial separable indefinite natural systems”, Journal of Geometry and Physics, 2014
38. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290
39. V. A. Sharafutdinov, “Killing tensor fields on the $2$-torus”, Siberian Math. J., 57:1 (2016), 155–173
40. V. V. Kozlov, D. V. Treschev, “Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics”, Sb. Math., 207:10 (2016), 1435–1449
41. I. A. Taimanov, “On first integrals of geodesic flows on a two-torus”, Proc. Steklov Inst. Math., 295 (2016), 225–242
42. S. V. Bolotin, V. V. Kozlov, “Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom”, Izv. Math., 81:4 (2017), 671–687
43. Bolsinov A. Matveev V.S. Miranda E. Tabachnikov S., “Open Problems, Questions and Challenges in Finite-Dimensional Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170430
44. V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Math., 83:6 (2019), 1137–1173
45. Agapov S. Valyuzhenich A., “Polynomial Integrals of Magnetic Geodesic Flows on the 2-Torus on Several Energy Levels”, Discret. Contin. Dyn. Syst., 39:11 (2019), 6565–6583
46. S. V. Agapov, “O pervykh integralakh dvumernykh geodezicheskikh potokov”, Sib. matem. zhurn., 61:4 (2020), 721–734
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