Integrability of magnetic geodesic flows

This thesis investigates some aspects of the integrability problem of a Hamiltonian system. The Hamiltonian system with Hamiltonian function H = Xn i,j=1 1 2gij(x1, . . . , xn)pipj , describes the geodesic flow of a Riemannian metric ds2 = Pn i,j=1 gij(x1, . . . , xn)dxidxj on an n-dimensional manifold. Some results from the research article, Polynomials integrals of magnetic geodesic flows on the 2-torus on several energy levels [3], are studied. In particular, a complex structure on the 2-torus is constructed to prove that if the geodesic flow with non-zero magnetic field on the 2-torus admits an additional cubic-in-momenta first integral on two different energy levels, then the magnetic field and the metric are functions of one variable.