January  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., ().   Google Scholar

[2]

M. Bialy, On periodic solutions for a reduction of Benney chain,, Nonlin. Diff. Eq. and Appl., 16 (2009), 731.   Google Scholar

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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.  Google Scholar

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M. Bialy, Integrable geodesic flows on surfaces,, Geom. and Funct. Analysis, 20 (2010), 357.  doi: doi:10.1007/s00039-010-0069-4.  Google Scholar

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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.  Google Scholar

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A. V. Bolsinov and A. T. Fomenko, "Integrable Geodesic Flows on two Dimensional Surfaces,", Monographs in Contemporary Mathematics, (2000).   Google Scholar

[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.   Google Scholar

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H. R. Dullin and V. Matveev, A new integrable system on the sphere,, Math Research Letters, 11 (2004), 715.   Google Scholar

[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.  Google Scholar

[10]

I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and related algebraic structures,, Funktsional. Anal, 13 (1979), 13.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

A. M. Perelomov, "Integrable Systems of Classical Mechanics and Lie Algebras," Vol. I,, Birkhauser Verlag, (1990).   Google Scholar

[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.  Google Scholar

[15]

D. Serre, "Systems of Conservation Laws," Vol. 2,, Geometric structures, (1996).   Google Scholar

[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.   Google Scholar

show all references

References:
[1]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," (Graduate Texts in Mathematics v. 60),, Springer., ().   Google Scholar

[2]

M. Bialy, On periodic solutions for a reduction of Benney chain,, Nonlin. Diff. Eq. and Appl., 16 (2009), 731.   Google Scholar

[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.  Google Scholar

[4]

M. Bialy, Integrable geodesic flows on surfaces,, Geom. and Funct. Analysis, 20 (2010), 357.  doi: doi:10.1007/s00039-010-0069-4.  Google Scholar

[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.  Google Scholar

[6]

A. V. Bolsinov and A. T. Fomenko, "Integrable Geodesic Flows on two Dimensional Surfaces,", Monographs in Contemporary Mathematics, (2000).   Google Scholar

[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.   Google Scholar

[8]

H. R. Dullin and V. Matveev, A new integrable system on the sphere,, Math Research Letters, 11 (2004), 715.   Google Scholar

[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.  Google Scholar

[10]

I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and related algebraic structures,, Funktsional. Anal, 13 (1979), 13.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

A. M. Perelomov, "Integrable Systems of Classical Mechanics and Lie Algebras," Vol. I,, Birkhauser Verlag, (1990).   Google Scholar

[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.  Google Scholar

[15]

D. Serre, "Systems of Conservation Laws," Vol. 2,, Geometric structures, (1996).   Google Scholar

[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.   Google Scholar

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