Integrable Magnetic Geodesic Flows on 2-torus and Global Solutions of Quasi-linear Systems of PDEs
发布时间：2019年11月19日
浏览次数：6445
发布者: He Liu
主讲人： Andrey Mironov (University of Novosibirsk)
活动时间： 从 2019-11-21 14:00 到 16:00
场地： 北京国际数学研究中心，全斋全29教室
The only one example has been known of magnetic geodesic flow on the 2-torus which has a polynomial in momenta integral independent of the Hamiltonian. In this example the integral is linear in momenta and corresponds to a one parametric group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on one energy level is considered. This problem can be reduced to a remarkable Semi-hamiltonian system of quasi-linear PDEs and to the question of existence of smooth periodic solutions for this system. Our main result states that the pair of Liouville metric with zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of a quadratic in momenta integral. Thus our construction gives a new example of smooth periodic solution to the Semi-hamiltonian (Rich) quasi-linear system of PDEs. Results were obtained with M. Bialy (Tel Aviv) and S. Agapov (Novosibirsk).
