The long-time behaviour of solutions to the Vlasov—Poisson equation has been a topic of interest for many years. A particular situation which has been studied recently concerns solutions around an attractive point mass. This model is appealing since the interaction between particles and point mass can be computed explicitly, in fact, this is exactly the Kepler problem. A reoccurring tool will therefore be action-angle variables. With this the Kepler dynamics splits essentially in two cases: hyperbolic and elliptic trajectories.

First, we discuss the long-time behaviour of solutions to the Vlasov—Poisson equation, when supported on hyperbolic trajectories. Then, we give an overview of known results as well as the main difficulties of the analysis of solutions supported on elliptic trajectories.