Communications in Mathematical Physics

, Volume 351, Issue 3, pp 993–1007 | Cite as

Integrable Magnetic Geodesic Flows on 2-Torus: New Examples via Quasi-Linear System of PDEs



For a magnetic geodesic flow on the 2-torus the only known integrable example is that of a flow integrable for all energy levels. It has an integral linear in momenta and corresponds to a one parameter group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on a single energy level is considered. Then, in addition to the example mentioned above, a few other explicit examples with quadratic in momenta integrals can be constructed by means of the Maupertuis’ principle. Recently we proved that such an integrability problem can be reduced to a remarkable semi-Hamiltonian system of quasi-linear PDEs and to the question of the existence of smooth periodic solutions for this system. Our main result of the present paper states that any Liouville metric with the zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with a small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of an integral quadratic in momenta.


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  1. 1.
    Agapov, S.V.: On the integrable magnetic geodesic flow on a 2-torus. Sib. Electron. Math. Rep. 12, 868–873 (2015) (Russian)Google Scholar
  2. 2.
    Bialy M.L.: Rigidity for periodic magnetic fields. Ergod. Theor. Dyn. Syst. 20(6), 1619–1626 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bialy M.L.: On periodic solutions for a reduction of Benney chain. Nonlinear Differ. Equ. Appl. 16, 731–743 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bialy M.L., Mironov A.E.: New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces. Cent. Eur. J. Math. 10(5), 1596–1604 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bialy M.L., Mironov A.E.: Rich quasi-linear system for integrable geodesic flow on 2-torus. Discrete Contin. Dyn. Syst. Ser. A 29(1), 81–90 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bialy M.L., Mironov A.E.: Integrable geodesic flows on 2-torus: formal solutions and variational principle. J. Geom. Phys. 87(1), 39–47 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bialy M.L., Mironov A.E.: Cubic and quartic integrals for geodesic flow on 2-torus via a system of the hydrodynamic type. Nonlinearity 24(12), 3541–3554 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bialy M.L., Mironov A.E.: From polynomial integrals of Hamiltonian flows to a model of non-linear elasticity. J. Differ. Equ. 255(10), 3434–3446 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Birkhoff G.D.: Dynamical Systems, Vol. 9. American Mathematical Society Colloquium Publications, New York (1927)CrossRefGoogle Scholar
  10. 10.
    Bolotin, S.V.: First integrals of systems with gyroscopic forces. Vestn. Mosk. U. Mat. M 6, 75–82 (1984) (Russian)Google Scholar
  11. 11.
    Bolsinov A.V., Kozlov V.V., Fomenko A.T.: The Maupertuis principle and geodesic flows on a sphere arising from integrable cases in the dynamics of a rigid body. Russ. Math. Surv. 50(3), 473–501 (1995)CrossRefMATHGoogle Scholar
  12. 12.
    Bolsinov A.V., Jovanovic B.: Magnetic geodesic flows on coadjoint orbits. J. Phys. A Math. 39(16), 247–252 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Burns K., Matveev V.S.: On the rigidity of magnetic systems with the same magnetic geodesics. Proc. Am. Math. Soc 134(2), 427–434 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Denisova N.V., Kozlov V.V.: Polynomial integrals of geodesic flows on a two-dimensional torus. Russ. Acad. Sci. Sbornik Math. 83(2), 469–481 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dorizzi B., Grammaticos B., Ramani A., Winternitz P.: Integrable Hamiltonian systems with velocity–dependent potentials. J. Math. Phys. 26(12), 3070–3079 (1985)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Efimov D.I.: The magnetic geodesic flow on a homogeneous symplectic manifold. Sib. Math. J. 46(1), 83–93 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ferapontov E.V., Fordy A.P.: Non-homogeneous systems of hydrodynamic type, related to quadratic Hamiltonians with electromagnetic term. Physica D 108, 350–364 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Greenberg J.M., Rascle M.: Time-periodic solutions to systems of conservation laws. Arch. Ration. Mech. Anal. 115(4), 395–407 (1991)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    John F.: Partial Differential Equations. Reprint of the Fourth Edition. Applied Mathematical Sciences, 1. Springer, New York (1991)Google Scholar
  20. 20.
    Kolokol’tsov V.N.: Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities. Math. USSR Izv 21(2), 291–306 (1983)CrossRefMATHGoogle Scholar
  21. 21.
    Kozlov V.V.: Symmetries, Topology, and Resonances in Hamiltonian Mechanics. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  22. 22.
    Kozlov V.V., Treschev D.V.: On the integrability of Hamiltonian systems with toral position space. Math. USSR Sbornik 63(1), 121–139 (1989)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Marikhin V.G., Sokolov V.V.: Pairs of commuting Hamiltonians that are quadratic in momenta. Theoret. Math. Phys. 149(2), 1425–1436 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pavlov M.V., Tsarev S.P.: Tri-Hamiltonian structures of Egorov Systems of hydrodynamic type. Funct. Anal. Appl. 37(1), 32–45 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Serre, D.: Richness and the Classification of Quasi-Linear Hyperbolic Systems. Preprint IMA N597 (1989)Google Scholar
  26. 26.
    Sevennec, B.: Geometrie des systemes de lois de conservation. Mem. Soc. Math. France Marseille 56 (1994)Google Scholar
  27. 27.
    Taimanov I.A.: On an integrable magnetic geodesic flow on the two-torus. Regul. Chaotic Dyn. 20(6), 667–678 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ten V.V.: Polynomial first integrals for systems with gyroscopic forces. Math. Notes 68(1), 135–138 (2000)MathSciNetMATHGoogle Scholar
  29. 29.
    Tsarev S.P.: On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Dokl. Math. 31, 488–491 (1985)MATHGoogle Scholar
  30. 30.
    Tsarev S.P.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Math. USSR Izv 37(2), 397–419 (1991)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.School of Mathematical Sciences, Faculty of Exact Sciences Raymond and Beverly SacklerTel Aviv UniversityTel AvivIsrael

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