Hamiltonian systems give a very good description of those physical phenomena where the energy is (approximately) conserved: from planetary orbits to the motion of particles.

Typically, however, the dynamics is highly sensitive to the initial conditions and therefore it is difficult to find specific orbits in the systems such as those connecting two subsets of phase space or those which are periodic.

In the last 50 years, the least action principle and related variational methods have been proven extremely effective to detect the existence of such orbits in these unstable landscape since these methods exploit the robust topological features of phase space. On the other hand, it is usually a challenging and fascinating mathematical problem to understand when these methods converge and produce the desired orbit.

The concrete goal of this course is to give an introduction to variational methods for detecting periodic orbits in finite-dimensional systems. In the first part, we will look at systems in Euclidean space and we will see that the convexity of the Hamiltonian plays a crucial role. In the second part, we will move on to analyze systems on manifolds keeping as central example the motion of a particle in a magnetic field. If time permits, we will focus on the case of surfaces or on the case of small perturbations of so-called Zoll systems where more precise information about periodic orbits can be obtained.

What you will learn

1) To use the language of Hamiltonian and Lagrangian Dynamics on simple symplectic manifolds such as cotangent bundles.

2) To combine analytical and topological tools for showing existence of critical points of nonlinear functionals

3) To know the general landscape of existence results for periodic orbits and recognize it in the example of a particle in a magnetic field.

Prerequisites

Basic concepts in Functional Analysis and in the Theory of Manifolds.

Literature

A. Abbondandolo, Lectures on the free period Lagrangian action functional, J. Fixed Point Theory Appl., 13, 397–430 (2013).

A. Abbondandolo and G. Benedetti, On the local systolic optimality of Zoll contact forms, arXiv:1912.04187, preprint, 2019.

L. Asselle and M. Mazzucchelli, On Tonelli periodic orbits with low energy on surfaces, Trans. Amer. Math. Soc., 371 no. 5, 3001-3048 (2019).

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birhäuser, 2011.