  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. $f=\left|\alpha\right|^2$ 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 $d\geq3$ dimensions. Finally, we adapt the
classical globality results for $d=3$ to the generalized system.
