TY - JOUR

T1 - Integrable Magnetic Geodesic Flows on 2-Torus

T2 - New Examples via Quasi-Linear System of PDEs

AU - Agapov, S. V.

AU - Bialy, M.

AU - Mironov, A. E.

N1 - Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - For a magnetic geodesic flow on the 2-torus the only known integrable example is that of a flow integrable for all energy levels. It has an integral linear in momenta and corresponds to a one parameter group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on a single energy level is considered. Then, in addition to the example mentioned above, a few other explicit examples with quadratic in momenta integrals can be constructed by means of the Maupertuis’ principle. Recently we proved that such an integrability problem can be reduced to a remarkable semi-Hamiltonian system of quasi-linear PDEs and to the question of the existence of smooth periodic solutions for this system. Our main result of the present paper states that any Liouville metric with the zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with a small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of an integral quadratic in momenta.

AB - For a magnetic geodesic flow on the 2-torus the only known integrable example is that of a flow integrable for all energy levels. It has an integral linear in momenta and corresponds to a one parameter group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on a single energy level is considered. Then, in addition to the example mentioned above, a few other explicit examples with quadratic in momenta integrals can be constructed by means of the Maupertuis’ principle. Recently we proved that such an integrability problem can be reduced to a remarkable semi-Hamiltonian system of quasi-linear PDEs and to the question of the existence of smooth periodic solutions for this system. Our main result of the present paper states that any Liouville metric with the zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with a small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of an integral quadratic in momenta.

UR - http://www.scopus.com/inward/record.url?scp=85008472890&partnerID=8YFLogxK

U2 - 10.1007/s00220-016-2822-5

DO - 10.1007/s00220-016-2822-5

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AN - SCOPUS:85008472890

SN - 0010-3616

VL - 351

SP - 993

EP - 1007

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -