  We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive
new time splitting methods based on the decomposition of the Hamiltonian
functional between the kinetic and electric energy. Assuming smoothness of the
solutions, we study the order conditions of such methods. It appears that these
conditions are of Runge-Kutta-Nystr{\"o}m type. In the one dimensional case,
the order conditions can be further simplified, and efficient methods of order
6 with a reduced number of stages can be constructed. In the general case,
high-order methods can also be constructed using explicit computations of
commutators. Numerical results are performed and show the benefit of using
high-order splitting schemes in that context. Complete and self-contained
proofs of convergence results and rigorous error estimates are also given.
