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 and Continuous Dynamical Systems, 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531

[2]

Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543

[3]

Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045

[4]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977

[5]

Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2829-2840. doi: 10.3934/dcdss.2020121

[6]

Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135

[7]

Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079

[8]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158

[9]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581

[10]

Serge Nicaise, Simon Stingelin, Fredi Tröltzsch. Optimal control of magnetic fields in flow measurement. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 579-605. doi: 10.3934/dcdss.2015.8.579

[11]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577

[12]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173

[13]

Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104.

[14]

Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 933-942. doi: 10.3934/dcdsb.2012.17.933

[15]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643

[16]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148

[17]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390

[18]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[19]

Dubi Kelmer, Hee Oh. Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds. Journal of Modern Dynamics, 2021, 17: 401-434. doi: 10.3934/jmd.2021014

[20]

Gabriela P. Ovando. The geodesic flow on nilpotent Lie groups of steps two and three. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 327-352. doi: 10.3934/dcds.2021119

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (204)
  • HTML views (120)
  • Cited by (2)

Other articles
by authors

[Back to Top]