From mechanics and symmetries: An approach to superintegrability

Superintegrable Hamiltonian systems possess remarkable mathematical properties. Among others, for maximal superintegrability, if the trajectory is finite, then the motion will be closed and periodic. The Hamiltonian formalism allows us to completely characterize the dynamics of a physical system with one conservation law, the Hamiltonian, using a symplectic geometry. By imposing more integrals of motion than the number of dimensions, we are looking to classify “Natural” Hamiltonians possessing a magnetic field leading to superintegrability and how to linearize them. We will discuss applications of superintegrability, and recent results, after a more historic approach.