Mathematics

Symplectic diffeomorphisms constitute the group of symmetries of Hamiltonian systems in classical mechanics. For this reason, the geometry associated with these diffeomorphisms has been the subject of intense study in the last decades. One of the breakthroughs in the field was the construction of so-called capacities, that is, symplectic invariants measuring the size of sets in phase space. Many natural questions arise in this context: Are symplectic capacities unique? Is there any relationship between the capacity of a set and its volume? We shall see how such questions are related to Hamiltonian systems having a fully periodic flow, and discuss how to construct such flows in the realm of geodesic and magnetic systems, using a Nash-Moser implicit function theorem.