Lecture notes on closed orbits for twisted autonomous Tonelli Lagrangian\n flows

These notes were prepared in occasion of a mini-course given by the author at\nthe "CIMPA Research School - Hamiltonian and Lagrangian Dynamics" (10-19 March\n2015 - Salto, Uruguay). The talks were meant as an introduction to the problem\nof finding periodic orbits of prescribed energy for autonomous Tonelli\nLagrangian systems on the twisted cotangent bundle of a closed manifold. In the\nfirst part of the lecture notes, we put together in a general theorem old and\nnew results on the subject. In the second part, we focus on an important class\nof examples: magnetic flows on surfaces. For such systems, we discuss a special\nmethod, originally due to Taimanov, to find periodic orbits with low energy and\nwe study in detail the stability properties of the energy levels.\n