Conformally symplectic Dynamics

Dynamists have been studying Hamiltonian systems for a long time. However, many physical systems are dissipative and do not preserve a symplectic form. This is the case, for example, with systems involving friction, which multiply the symplectic form by a constant smaller than 1. We will prove that almost every point is in the unstable set of infinity for these systems and we will illustrate different situations that may arise with examples. We will also study invariant manifolds by such dynamics. We will provide an example where an invariant proper submanifold is not isotropic and give different conditions that imply that a given invariant submanifold is isotropic. In particular, we will outline a strange link between isotropy and entropy. Examples demonstrate that some systems have a global attractor, while others do not. We will give a sufficient condition for a conformally Hamiltonian dynamics of a cotangent bundle to have a global attractor. Then we will introduce two very classical examples : Ma$\tilde{\text{n}}$é example and damped mechanical systems. After that, we introduce the notion of locally symplectic manifold. Unlike symplectic manifolds, these manifolds carry conformally symplectic dynamics that have a conservative part and a dissipative part. For conformally Hamiltonian flow, we will describe dissipative and conservative orbits using their number of rotation. Notes of a course given at CIME school at Cetraro, june 2025