Single speed solutions of the Vlasov–Poisson equations for Coulombian and Newtonian systems in nD

Results of theoretical and numerical investigations concerning the space-time evolution in nD of Coulombian systems known in Plasma Physics and Newtonian systems known in Cosmology are reported here. A generic, canonical Hamiltonian formulation is given (1) for studying single speed solutions of their Coulomb- or Jeans–Vlasov–Poisson descriptions. (2) It is shown that their equations of motion are integrable for vorticity free planar motions in nD (3) and that their characteristics are solutions of explicit non-linear ODEs in general (4). Some simple examples of trajectories are illustrated and the problem of their crossing is evoked.