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 and Continuous Dynamical Systems, 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) 731-743.

[3]

M. Bialy, Polynomial integrals for a Hamiltonian system and breakdown of smooth solutions for quasi-linear equations, Nonlinearity, 7 (1994), 1169-1174. doi: doi:10.1088/0951-7715/7/4/005.

[4]

M. Bialy, Integrable geodesic flows on surfaces, Geom. and Funct. Analysis, 20 (2010), 357-367. 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), 925-930. 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, Consultants Bureau, New York, 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), 781-785; English transl. Soviet Math. Dokl., 27 (1983).

[8]

H. R. Dullin and V. Matveev, A new integrable system on the sphere, Math Research Letters, 11 (2004), 715-722.

[9]

H. R. Dullin, V. Matveev and P. Topalov, On integrals of third degree in momenta, Regular and Chaotic Dynamics, 4 (1999), 35-44. doi: doi:10.1070/rd1999v004n03ABEH000114.

[10]

I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and related algebraic structures, Funktsional. Anal, i Prilozhen., 13 (1979), 13-30.

[11]

L. S. Hall, A theory of exact and approximate configuration invariants, Physica D, 8 (1983), 90-116. 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), 994-1010.

[13]

A. M. Perelomov, "Integrable Systems of Classical Mechanics and Lie Algebras," Vol. I, Birkhauser Verlag, Basel, 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), 641-663. doi: doi:10.1007/s002200050740.

[15]

D. Serre, "Systems of Conservation Laws," Vol. 2, Geometric structures, oscillations, and initial-boundary value problems. Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 2000.

[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), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419.

show all references

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) 731-743.

[3]

M. Bialy, Polynomial integrals for a Hamiltonian system and breakdown of smooth solutions for quasi-linear equations, Nonlinearity, 7 (1994), 1169-1174. doi: doi:10.1088/0951-7715/7/4/005.

[4]

M. Bialy, Integrable geodesic flows on surfaces, Geom. and Funct. Analysis, 20 (2010), 357-367. 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), 925-930. 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, Consultants Bureau, New York, 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), 781-785; English transl. Soviet Math. Dokl., 27 (1983).

[8]

H. R. Dullin and V. Matveev, A new integrable system on the sphere, Math Research Letters, 11 (2004), 715-722.

[9]

H. R. Dullin, V. Matveev and P. Topalov, On integrals of third degree in momenta, Regular and Chaotic Dynamics, 4 (1999), 35-44. doi: doi:10.1070/rd1999v004n03ABEH000114.

[10]

I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and related algebraic structures, Funktsional. Anal, i Prilozhen., 13 (1979), 13-30.

[11]

L. S. Hall, A theory of exact and approximate configuration invariants, Physica D, 8 (1983), 90-116. 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), 994-1010.

[13]

A. M. Perelomov, "Integrable Systems of Classical Mechanics and Lie Algebras," Vol. I, Birkhauser Verlag, Basel, 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), 641-663. doi: doi:10.1007/s002200050740.

[15]

D. Serre, "Systems of Conservation Laws," Vol. 2, Geometric structures, oscillations, and initial-boundary value problems. Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 2000.

[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), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419.

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