Generalized symplectization of Vlasov dynamics and application to the Vlasov-Poisson system

In this paper, we study a Hamiltonian structure of the Vlasov-Poisson system, first mentioned by Fr"ohlich, Knowles, and Schwarz. To begin with, we give a formal guideline to derive a Hamiltonian on a subspace of complex-valued $L^2$ integrable functions $\alpha$ on the one particle phase space ${\mathbb{R}}^{2d}$, s.t. $f={\mid \alpha \mid }^2$ is a solution of a collisionless Boltzmann equation. The only requirement is a sufficiently regular energy functional on a subspace of distribution functions $f\in L^1$. Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov-Poisson in $d\ge 3$ dimensions. Finally, we adapt the classical globality results for $d=3$ to the generalized system.