Agapov, S. V.; Bialy, M.; Mironov, A. E. Integrable magnetic geodesic flows on 2-torus: new examples via quasi-linear system of PDEs. (English) Zbl 1364.37123 Commun. Math. Phys. 351, No. 3, 993-1007 (2017). The main result states that Liouville metrics \([\Lambda_1(x)+\Lambda_2(y)](dx^2+dy^2)\) on the 2-torus admits paths of perturbations into analytic Riemannian metrics in such a way that there exist non-zero analytic magnetic fields, such that magnetic geodesic flows on one energy level have a first integral which is quadratic with respect to the momenta. This provides explicit examples of magnetic geodesic flows which are integrable on some isolated energy level. No such example seemed to be known before.The strategy of the proof is to reduce the statement to solving a matricial PDE having the following form: some \(U\equiv U(t,x,y)\in C^\infty ([0,\varepsilon[\times\mathbb T^2,\mathbb R^4)\) must solve \(\partial_tU=A(U)\partial_xU+B(U)\partial_yU\), for convenient matrices \(A(U)\), \(B(U)\in\mathcal M_4(\mathbb R)\).The authors consider then the magnetic flow of a Riemannian metric \(\Lambda(x,y)(dx^2+dy^2)\) on the 2-torus, with a non-zero magnetic form \(\omega\), such that the magnetic geodesic flow has a first integral \(F\) on all energy levels which is quadratic in momenta \((p_1,p_2)\).They show that a convenient choice of the coordinates reduces this case to \(\Lambda(x,y)=\Lambda(y)\), \(\omega=-u'(y)dx\wedge dy\), and that \(F\) is necessarily a linear combination of the Hamiltonian \(\frac{p^2_1+p^2_2}{2\Lambda(y)}\) and of the particular first integral \(F_0:=p_1+u(y)\). Reviewer: Jacques Franchi (Strasbourg) Cited in 10 Documents MSC: 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry Keywords:2-torus; Liouville metrics; magnetic geodesic flow; quadratic first integral × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Agapov, S.V.: On the integrable magnetic geodesic flow on a 2-torus. Sib. Electron. Math. Rep. 12, 868-873 (2015) (Russian) · Zbl 1347.35157 [2] Bialy M.L.: Rigidity for periodic magnetic fields. Ergod. Theor. Dyn. Syst. 20(6), 1619-1626 (2000) · Zbl 1003.37035 · doi:10.1017/S0143385700000894 [3] Bialy M.L.: On periodic solutions for a reduction of Benney chain. Nonlinear Differ. Equ. Appl. 16, 731-743 (2009) · Zbl 1182.35014 · doi:10.1007/s00030-009-0032-y [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) · Zbl 1259.35137 · doi:10.2478/s11533-012-0045-3 [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) · Zbl 1232.37035 · doi:10.3934/dcds.2011.29.81 [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) · Zbl 1304.53084 · doi:10.1016/j.geomphys.2014.08.006 [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) · Zbl 1232.35092 · doi:10.1088/0951-7715/24/12/010 [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) · Zbl 1320.35238 · doi:10.1016/j.jde.2013.07.040 [9] Birkhoff G.D.: Dynamical Systems, Vol. 9. American Mathematical Society Colloquium Publications, New York (1927) · JFM 53.0732.01 · doi:10.1090/coll/009 [10] Bolotin, S.V.: First integrals of systems with gyroscopic forces. Vestn. Mosk. U. Mat. M 6, 75-82 (1984) (Russian) · Zbl 0597.70019 [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) · Zbl 0881.58031 · doi:10.1070/RM1995v050n03ABEH002100 [12] Bolsinov A.V., Jovanovic B.: Magnetic geodesic flows on coadjoint orbits. J. Phys. A Math. 39(16), 247-252 (2006) · Zbl 1089.70010 · doi:10.1088/0305-4470/39/16/L01 [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) · Zbl 1079.37019 · doi:10.1090/S0002-9939-05-08196-7 [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) · Zbl 0841.53039 · doi:10.1070/SM1995v083n02ABEH003601 [15] Dorizzi B., Grammaticos B., Ramani A., Winternitz P.: Integrable Hamiltonian systems with velocity-dependent potentials. J. Math. Phys. 26(12), 3070-3079 (1985) · Zbl 0586.70013 · doi:10.1063/1.526685 [16] Efimov D.I.: The magnetic geodesic flow on a homogeneous symplectic manifold. Sib. Math. J. 46(1), 83-93 (2005) · Zbl 1079.37055 · doi:10.1007/s11202-005-0009-y [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) · Zbl 0933.37061 · doi:10.1016/S0167-2789(97)00040-7 [18] Greenberg J.M., Rascle M.: Time-periodic solutions to systems of conservation laws. Arch. Ration. Mech. Anal. 115(4), 395-407 (1991) · Zbl 0769.35037 · doi:10.1007/BF00375281 [19] John F.: Partial Differential Equations. Reprint of the Fourth Edition. Applied Mathematical Sciences, 1. Springer, New York (1991) [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) · Zbl 0548.58028 · doi:10.1070/IM1983v021n02ABEH001792 [21] Kozlov V.V.: Symmetries, Topology, and Resonances in Hamiltonian Mechanics. Springer, Berlin (1996) · Zbl 0921.58029 · doi:10.1007/978-3-642-78393-7 [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) · Zbl 0696.58022 · doi:10.1070/SM1989v063n01ABEH003263 [23] Marikhin V.G., Sokolov V.V.: Pairs of commuting Hamiltonians that are quadratic in momenta. Theoret. Math. Phys. 149(2), 1425-1436 (2006) · Zbl 1177.70021 · doi:10.1007/s11232-006-0129-y [24] Pavlov M.V., Tsarev S.P.: Tri-Hamiltonian structures of Egorov Systems of hydrodynamic type. Funct. Anal. Appl. 37(1), 32-45 (2003) · Zbl 1019.37048 · doi:10.1023/A:1022971910438 [25] Serre, D.: Richness and the Classification of Quasi-Linear Hyperbolic Systems. Preprint IMA N597 (1989) [26] Sevennec, B.: Geometrie des systemes de lois de conservation. Mem. Soc. Math. France Marseille 56 (1994) · Zbl 0807.35090 [27] Taimanov I.A.: On an integrable magnetic geodesic flow on the two-torus. Regul. Chaotic Dyn. 20(6), 667-678 (2015) · Zbl 1342.53108 · doi:10.1134/S1560354715060039 [28] Ten V.V.: Polynomial first integrals for systems with gyroscopic forces. Math. Notes 68(1), 135-138 (2000) · Zbl 0995.37043 [29] Tsarev S.P.: On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Dokl. Math. 31, 488-491 (1985) · Zbl 0605.35075 [30] Tsarev S.P.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Math. USSR Izv 37(2), 397-419 (1991) · Zbl 0796.76014 · doi:10.1070/IM1991v037n02ABEH002069 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.