# American Institute of Mathematical Sciences

2011, 29(1): 81-90. doi: 10.3934/dcds.2011.29.81

## Rich quasi-linear system for integrable geodesic flows on 2-torus

 1 School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel 2 Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, 630090, Russian Federation

Received  October 2009 Revised  April 2010 Published  September 2010

Consider a Riemannian metric on two-torus. We prove that the question of existence of polynomial first integrals leads naturally to a remarkable system of quasi-linear equations which turns out to be a Rich system of conservation laws. This reduces the question of integrability to the question of existence of smooth (quasi-) periodic solutions for this Rich quasi-linear system.
Citation: Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," (Graduate Texts in Mathematics v. 60),, Springer., (). [2] M. Bialy, On periodic solutions for a reduction of Benney chain,, Nonlin. Diff. Eq. and Appl., 16 (2009), 731. [3] M. Bialy, Polynomial integrals for a Hamiltonian system and breakdown of smooth solutions for quasi-linear equations,, Nonlinearity, 7 (1994), 1169. doi: doi:10.1088/0951-7715/7/4/005. [4] M. Bialy, Integrable geodesic flows on surfaces,, Geom. and Funct. Analysis, 20 (2010), 357. doi: doi:10.1007/s00039-010-0069-4. [5] M. Bialy, Hamiltonian form and infinity many conservation laws for a quasiliner system,, Nonlinearity, 10 (1997), 925. doi: doi:10.1088/0951-7715/10/4/007. [6] A. V. Bolsinov and A. T. Fomenko, "Integrable Geodesic Flows on two Dimensional Surfaces,", Monographs in Contemporary Mathematics, (2000). [7] B. A. Dubrovin and S. P. Novikov, Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov- Whitham averaging method,, Dokl. Akad. Nauk SSSR, 270 (1983), 781. [8] H. R. Dullin and V. Matveev, A new integrable system on the sphere,, Math Research Letters, 11 (2004), 715. [9] H. R. Dullin, V. Matveev and P. Topalov, On integrals of third degree in momenta,, Regular and Chaotic Dynamics, 4 (1999), 35. doi: doi:10.1070/rd1999v004n03ABEH000114. [10] I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and related algebraic structures,, Funktsional. Anal, 13 (1979), 13. [11] L. S. Hall, A theory of exact and approximate configuration invariants,, Physica D, 8 (1983), 90. doi: doi:10.1016/0167-2789(83)90312-3. [12] 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 (1982), 994. [13] A. M. Perelomov, "Integrable Systems of Classical Mechanics and Lie Algebras," Vol. I,, Birkhauser Verlag, (1990). [14] E. N. Selivanova, New examples of Integrable Conservative Systems on $S^2$ and the Case of Goryachev-Chaplygin,, Comm. Math. Phys., 207 (1999), 641. doi: doi:10.1007/s002200050740. [15] D. Serre, "Systems of Conservation Laws," Vol. 2,, Geometric structures, (1996). [16] S. P. Tsarev, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048.

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##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," (Graduate Texts in Mathematics v. 60),, Springer., (). [2] M. Bialy, On periodic solutions for a reduction of Benney chain,, Nonlin. Diff. Eq. and Appl., 16 (2009), 731. [3] M. Bialy, Polynomial integrals for a Hamiltonian system and breakdown of smooth solutions for quasi-linear equations,, Nonlinearity, 7 (1994), 1169. doi: doi:10.1088/0951-7715/7/4/005. [4] M. Bialy, Integrable geodesic flows on surfaces,, Geom. and Funct. Analysis, 20 (2010), 357. doi: doi:10.1007/s00039-010-0069-4. [5] M. Bialy, Hamiltonian form and infinity many conservation laws for a quasiliner system,, Nonlinearity, 10 (1997), 925. doi: doi:10.1088/0951-7715/10/4/007. [6] A. V. Bolsinov and A. T. Fomenko, "Integrable Geodesic Flows on two Dimensional Surfaces,", Monographs in Contemporary Mathematics, (2000). [7] B. A. Dubrovin and S. P. Novikov, Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov- Whitham averaging method,, Dokl. Akad. Nauk SSSR, 270 (1983), 781. [8] H. R. Dullin and V. Matveev, A new integrable system on the sphere,, Math Research Letters, 11 (2004), 715. [9] H. R. Dullin, V. Matveev and P. Topalov, On integrals of third degree in momenta,, Regular and Chaotic Dynamics, 4 (1999), 35. doi: doi:10.1070/rd1999v004n03ABEH000114. [10] I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and related algebraic structures,, Funktsional. Anal, 13 (1979), 13. [11] L. S. Hall, A theory of exact and approximate configuration invariants,, Physica D, 8 (1983), 90. doi: doi:10.1016/0167-2789(83)90312-3. [12] 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 (1982), 994. [13] A. M. Perelomov, "Integrable Systems of Classical Mechanics and Lie Algebras," Vol. I,, Birkhauser Verlag, (1990). [14] E. N. Selivanova, New examples of Integrable Conservative Systems on $S^2$ and the Case of Goryachev-Chaplygin,, Comm. Math. Phys., 207 (1999), 641. doi: doi:10.1007/s002200050740. [15] D. Serre, "Systems of Conservation Laws," Vol. 2,, Geometric structures, (1996). [16] S. P. Tsarev, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048.
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