New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces

Misha Bialy; Andrey Mironov

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1596-1604
  • ISSN: 2391-5455

Abstract

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We consider magnetic geodesic flows on the two-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.

How to cite

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Misha Bialy, and Andrey Mironov. "New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces." Open Mathematics 10.5 (2012): 1596-1604. <http://eudml.org/doc/269054>.

@article{MishaBialy2012,
abstract = {We consider magnetic geodesic flows on the two-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.},
author = {Misha Bialy, Andrey Mironov},
journal = {Open Mathematics},
keywords = {Integral of motion; Magnetic geodesic flows; Riemann invariants; Systems of hydrodynamic type; integral of motion; magnetic geodesic flows; systems of hydrodynamic type; polynomial in momenta first integrals},
language = {eng},
number = {5},
pages = {1596-1604},
title = {New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces},
url = {http://eudml.org/doc/269054},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Misha Bialy
AU - Andrey Mironov
TI - New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1596
EP - 1604
AB - We consider magnetic geodesic flows on the two-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.
LA - eng
KW - Integral of motion; Magnetic geodesic flows; Riemann invariants; Systems of hydrodynamic type; integral of motion; magnetic geodesic flows; systems of hydrodynamic type; polynomial in momenta first integrals
UR - http://eudml.org/doc/269054
ER -

References

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  1. [1] Bialy M., On periodic solutions for a reduction of Benney chain, NoDEA Nonlinear Differential Equations Appl., 2009, 16(6), 731–743 http://dx.doi.org/10.1007/s00030-009-0032-y Zbl1182.35014
  2. [2] Bialy M., Integrable geodesic flows on surfaces, Geom. Funct. Anal., 2010, 20(2), 357–367 http://dx.doi.org/10.1007/s00039-010-0069-4 Zbl1203.37092
  3. [3] Bialy M., Richness or semi-Hamiltonicity of quasi-linear systems which are not in evolution form, preprint available at http://arxiv.org/abs/1101.5897 
  4. [4] Bialy M., Mironov A.E., Rich quasi-linear system for integrable geodesic flows on 2-torus, Discrete Contin. Dyn. Syst., 2011, 29(1), 81–90 http://dx.doi.org/10.3934/dcds.2011.29.81 Zbl1232.37035
  5. [5] Bialy M., Mironov A.E., Cubic and quartic integrals for geodesic flow on 2-torus via system of hydrodynamic type, Nonlinearity, 2011, 24(12), 3541–3554 http://dx.doi.org/10.1088/0951-7715/24/12/010 Zbl1232.35092
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  10. [10] Mokhov O.I., Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature, Russian Math. Surveys, 1990, 45(3), 218–219 http://dx.doi.org/10.1070/RM1990v045n03ABEH002351 Zbl0712.35080
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  12. [12] Sévennec B., Géométrie des Systèmes Hyperboliques de Lois de Conservation, Mém. Soc. Math. France (N.S.), 56, Société Mathématique de France, Marseille, 1994 
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