November  2019, 39(11): 6565-6583. doi: 10.3934/dcds.2019285

Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels

1. 

Sobolev Institute of Mathematics, Novosibirsk, 4 Acad. Koptyug avenue, 630090, Russia

2. 

Novosibirsk State University, Novosibirsk, 1 Pirogova str., 630090, Russia

* Corresponding author: Sergei Agapov

Received  January 2019 Published  August 2019

Fund Project: The first author is supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation)

In this paper the geodesic flow on the 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral $ F $ on $ N+2 $ different energy levels which is polynomial in momenta of an arbitrary degree $ N $ with analytic periodic coefficients. It is proved that in this case the magnetic field and the metric are functions of one variable and there exists a linear in momenta first integral on all energy levels.

Citation: Sergei Agapov, Alexandr Valyuzhenich. Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6565-6583. doi: 10.3934/dcds.2019285
References:
[1]

S. V. Agapov and D. N. Aleksandrov, Fourth-degree polynomial integrals of a natural mechanical system on a two-dimensional torus, Math. Notes, 93 (2013), 780-783. doi: 10.1134/S0001434613050155. Google Scholar

[2]

S. V. AgapovM. Bialy and A. E. Mironov, Integrable magnetic geodesic flows on 2-torus: New examples via quasi-linear system of PDEs, Comm. Math. Phys., 351 (2017), 993-1007. doi: 10.1007/s00220-016-2822-5. Google Scholar

[3]

M. L. , First integrals that are polynomial in momenta for a mechanical system on a two-dimensional torus, Funct. Anal. Appl., 21 (1987), 64-65. doi: 10.1007/BF01077805. Google Scholar

[4]

M. L. Bialy, Rigidity for periodic magnetic fields, Theor. Dyn. Syst., 20 (2000), 1619-1626. doi: 10.1017/S0143385700000894. Google Scholar

[5]

M. Bialy and A. E. Mironov, Rich quasi-linear system for integrable geodesic flow on 2-torus, Discrete Continuous Dynam. Systems-A, 29 (2011), 81-90. doi: 10.3934/dcds.2011.29.81. Google Scholar

[6]

M. L. Bialy and A. E. Mironov, Integrable geodesic flows on 2-torus: Formal solutions and variational principle, Journal of Geometry and Physics, 87 (2015), 39-47. doi: 10.1016/j.geomphys.2014.08.006. Google Scholar

[7]

M. Bialy and A. E. Mironov, Cubic and quartic integrals for geodesic flow on 2-torus via a system of the hydrodynamic type, Nonlinearity, 24 (2011), 3541-3554. doi: 10.1088/0951-7715/24/12/010. Google Scholar

[8]

M. Bialy and A. Mironov, New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces, Cent. Eur. J. Math., 10 (2012), 1596-1604. doi: 10.2478/s11533-012-0045-3. Google Scholar

[9]

S. V. Bolotin, First integrals of systems with gyroscopic forces, Vestn. Mosk. U. Mat. M., (1984), 75–82,113. Google Scholar

[10]

A. V. BolsinovV. V. Kozlov and A. T. Fomenko, The Maupertuis principle and geodesic flows on a sphere arising from integrable cases in the dynamics of a rigid body, Russian Math. Surveys, 50 (1995), 473-501. doi: 10.1070/RM1995v050n03ABEH002100. Google Scholar

[11]

A. V. Bolsinov and B. Jovanović, Magnetic geodesic flows on coadjoint orbits, J. Phys. A-Math, 39 (2006), L247–L252. doi: 10.1088/0305-4470/39/16/L01. Google Scholar

[12]

K. Burns and V. S. Matveev, On the rigidity of magnetic systems with the same magnetic geodesics, P. Am. Math. Soc., 134 (2006), 427–434. Available from: http://www.ams.org/journals/proc/2006-134-02/S0002-9939-05-08196-7/S0002-9939-05-08196-7.pdf. doi: 10.1090/S0002-9939-05-08196-7. Google Scholar

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N. V. Denisova and V. V. Kozlov, Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space, Russian Acad. Sci. Sb. Math., 191 (2000), 189-208. doi: 10.1070/SM2000v191n02ABEH000452. Google Scholar

[14]

N. V. DenisovaV. V. Kozlov and D. V. Treschëv, Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space, Izv. Math., 76 (2012), 907-921. doi: 10.1070/IM2012v076n05ABEH002609. Google Scholar

[15]

B. DorizziB. GrammaticosA. Ramani and P. Winternitz, Integrable Hamiltonian systems with velocity-dependent potentials, J. Math. Phys., 26 (1985), 3070-3079. doi: 10.1063/1.526685. Google Scholar

[16]

D. I. Efimov, The magnetic geodesic flow in a homogeneous field on the complex projective space, Siberian Math. J., 45 (2004), 465-474. doi: 10.1023/B:SIMJ.0000028611.65071.bd. Google Scholar

[17]

D. I. Efimov, The magnetic geodesic flow on a homogeneous symplectic manifold, Siberian Math. J., 46 (2005), 83-93. doi: 10.1007/s11202-005-0009-y. Google Scholar

[18]

V. N. Kolokol'tsov, Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities, Math. USSR Izv., 21 (1983), 291-306. doi: 10.1070/IM1983v021n02ABEH001792. Google Scholar

[19]

V. V. Kozlov, Topological obstacles to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302. Google Scholar

[20]

V. V. Kozlov, Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Izdatel'stvo Udmurt-skogo Universiteta, Izhevsk, 1995. doi: 10.1007/978-3-642-78393-7. Google Scholar

[21]

V. V. Kozlov and N. V. Denisova, Symmetries and the topology of dynamical systems with two degrees of freedom, Russian Acad. Sci. Sb. Math., 80 (1995), 105-124. doi: 10.1070/SM1995v080n01ABEH003516. Google Scholar

[22]

V. V. Kozlov and N. V. Denisova, Polynomial integrals of geodesic flows on a two-dimensional torus, Russian Acad. Sci. Sb. Math., 83 (1995), 469-481. doi: 10.1070/SM1995v083n02ABEH003601. Google Scholar

[23]

V. V. Kozlov and D. V. Treschëv, On the integrability of Hamiltonian systems with toral position space, Math. USSR Sb., 63 (1989), 121-139. doi: 10.1070/SM1989v063n01ABEH003263. Google Scholar

[24]

A. E. Mironov, On polynomial integrals of a mechanical system on a two-dimensional torus, Izv. Math., 74 (2010), 805-817. doi: 10.1070/IM2010v074n04ABEH002508. Google Scholar

[25]

I. A. Taimanov, On an integrable magnetic geodesic flow on the two-torus, Regul. Chaotic Dyn., 20 (2015), 667-678. doi: 10.1134/S1560354715060039. Google Scholar

[26]

I. A. Taimanov, On first integrals of geodesic flows on a two-torus, Proc. Steklov Inst. Math., 295 (2016), 225-242. doi: 10.1134/S0081543816080150. Google Scholar

[27]

V. V. Ten, Polynomial first integrals for systems with gyroscopic forces, Math. Notes, 68 (2000), 135–138. Available from: https://link.springer.com/article/10.1007/BF02674658. Google Scholar

[28]

S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR-Izv, 37 (1991), 397-419. doi: 10.1070/IM1991v037n02ABEH002069. Google Scholar

show all references

References:
[1]

S. V. Agapov and D. N. Aleksandrov, Fourth-degree polynomial integrals of a natural mechanical system on a two-dimensional torus, Math. Notes, 93 (2013), 780-783. doi: 10.1134/S0001434613050155. Google Scholar

[2]

S. V. AgapovM. Bialy and A. E. Mironov, Integrable magnetic geodesic flows on 2-torus: New examples via quasi-linear system of PDEs, Comm. Math. Phys., 351 (2017), 993-1007. doi: 10.1007/s00220-016-2822-5. Google Scholar

[3]

M. L. , First integrals that are polynomial in momenta for a mechanical system on a two-dimensional torus, Funct. Anal. Appl., 21 (1987), 64-65. doi: 10.1007/BF01077805. Google Scholar

[4]

M. L. Bialy, Rigidity for periodic magnetic fields, Theor. Dyn. Syst., 20 (2000), 1619-1626. doi: 10.1017/S0143385700000894. Google Scholar

[5]

M. Bialy and A. E. Mironov, Rich quasi-linear system for integrable geodesic flow on 2-torus, Discrete Continuous Dynam. Systems-A, 29 (2011), 81-90. doi: 10.3934/dcds.2011.29.81. Google Scholar

[6]

M. L. Bialy and A. E. Mironov, Integrable geodesic flows on 2-torus: Formal solutions and variational principle, Journal of Geometry and Physics, 87 (2015), 39-47. doi: 10.1016/j.geomphys.2014.08.006. Google Scholar

[7]

M. Bialy and A. E. Mironov, Cubic and quartic integrals for geodesic flow on 2-torus via a system of the hydrodynamic type, Nonlinearity, 24 (2011), 3541-3554. doi: 10.1088/0951-7715/24/12/010. Google Scholar

[8]

M. Bialy and A. Mironov, New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces, Cent. Eur. J. Math., 10 (2012), 1596-1604. doi: 10.2478/s11533-012-0045-3. Google Scholar

[9]

S. V. Bolotin, First integrals of systems with gyroscopic forces, Vestn. Mosk. U. Mat. M., (1984), 75–82,113. Google Scholar

[10]

A. V. BolsinovV. V. Kozlov and A. T. Fomenko, The Maupertuis principle and geodesic flows on a sphere arising from integrable cases in the dynamics of a rigid body, Russian Math. Surveys, 50 (1995), 473-501. doi: 10.1070/RM1995v050n03ABEH002100. Google Scholar

[11]

A. V. Bolsinov and B. Jovanović, Magnetic geodesic flows on coadjoint orbits, J. Phys. A-Math, 39 (2006), L247–L252. doi: 10.1088/0305-4470/39/16/L01. Google Scholar

[12]

K. Burns and V. S. Matveev, On the rigidity of magnetic systems with the same magnetic geodesics, P. Am. Math. Soc., 134 (2006), 427–434. Available from: http://www.ams.org/journals/proc/2006-134-02/S0002-9939-05-08196-7/S0002-9939-05-08196-7.pdf. doi: 10.1090/S0002-9939-05-08196-7. Google Scholar

[13]

N. V. Denisova and V. V. Kozlov, Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space, Russian Acad. Sci. Sb. Math., 191 (2000), 189-208. doi: 10.1070/SM2000v191n02ABEH000452. Google Scholar

[14]

N. V. DenisovaV. V. Kozlov and D. V. Treschëv, Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space, Izv. Math., 76 (2012), 907-921. doi: 10.1070/IM2012v076n05ABEH002609. Google Scholar

[15]

B. DorizziB. GrammaticosA. Ramani and P. Winternitz, Integrable Hamiltonian systems with velocity-dependent potentials, J. Math. Phys., 26 (1985), 3070-3079. doi: 10.1063/1.526685. Google Scholar

[16]

D. I. Efimov, The magnetic geodesic flow in a homogeneous field on the complex projective space, Siberian Math. J., 45 (2004), 465-474. doi: 10.1023/B:SIMJ.0000028611.65071.bd. Google Scholar

[17]

D. I. Efimov, The magnetic geodesic flow on a homogeneous symplectic manifold, Siberian Math. J., 46 (2005), 83-93. doi: 10.1007/s11202-005-0009-y. Google Scholar

[18]

V. N. Kolokol'tsov, Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities, Math. USSR Izv., 21 (1983), 291-306. doi: 10.1070/IM1983v021n02ABEH001792. Google Scholar

[19]

V. V. Kozlov, Topological obstacles to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302. Google Scholar

[20]

V. V. Kozlov, Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Izdatel'stvo Udmurt-skogo Universiteta, Izhevsk, 1995. doi: 10.1007/978-3-642-78393-7. Google Scholar

[21]

V. V. Kozlov and N. V. Denisova, Symmetries and the topology of dynamical systems with two degrees of freedom, Russian Acad. Sci. Sb. Math., 80 (1995), 105-124. doi: 10.1070/SM1995v080n01ABEH003516. Google Scholar

[22]

V. V. Kozlov and N. V. Denisova, Polynomial integrals of geodesic flows on a two-dimensional torus, Russian Acad. Sci. Sb. Math., 83 (1995), 469-481. doi: 10.1070/SM1995v083n02ABEH003601. Google Scholar

[23]

V. V. Kozlov and D. V. Treschëv, On the integrability of Hamiltonian systems with toral position space, Math. USSR Sb., 63 (1989), 121-139. doi: 10.1070/SM1989v063n01ABEH003263. Google Scholar

[24]

A. E. Mironov, On polynomial integrals of a mechanical system on a two-dimensional torus, Izv. Math., 74 (2010), 805-817. doi: 10.1070/IM2010v074n04ABEH002508. Google Scholar

[25]

I. A. Taimanov, On an integrable magnetic geodesic flow on the two-torus, Regul. Chaotic Dyn., 20 (2015), 667-678. doi: 10.1134/S1560354715060039. Google Scholar

[26]

I. A. Taimanov, On first integrals of geodesic flows on a two-torus, Proc. Steklov Inst. Math., 295 (2016), 225-242. doi: 10.1134/S0081543816080150. Google Scholar

[27]

V. V. Ten, Polynomial first integrals for systems with gyroscopic forces, Math. Notes, 68 (2000), 135–138. Available from: https://link.springer.com/article/10.1007/BF02674658. Google Scholar

[28]

S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR-Izv, 37 (1991), 397-419. doi: 10.1070/IM1991v037n02ABEH002069. Google Scholar

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