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.

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.

[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.

[1]

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

[2]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

[3]

John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7.

[4]

Daniel Visscher. A new proof of Franks' lemma for geodesic flows. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4875-4895. doi: 10.3934/dcds.2014.34.4875

[5]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841

[6]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147

[7]

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

[8]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[9]

Jong Uhn Kim. On the stochastic Burgers equation with a polynomial nonlinearity in the real line. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 835-866. doi: 10.3934/dcdsb.2006.6.835

[10]

Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403

[11]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477

[12]

Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61

[13]

Jeffrey Boland. On rigidity properties of contact time changes of locally symmetric geodesic flows. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 645-650. doi: 10.3934/dcds.2000.6.645

[14]

Igor Kossowski, Katarzyna Szymańska-Dębowska. Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 275-281. doi: 10.3934/dcdsb.2018019

[15]

Alexander I. Bufetov. Hölder cocycles and ergodic integrals for translation flows on flat surfaces. Electronic Research Announcements, 2010, 17: 34-42. doi: 10.3934/era.2010.17.34

[16]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683

[17]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[18]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[19]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[20]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]