  We consider Hamiltonian closures of the Vlasov equation using the phase-space
moments of the distribution function. We provide some conditions on the
closures imposed by the Jacobi identity. We completely solve some families of
examples. As a result, we show that imposing that the resulting reduced system
preserves the Hamiltonian character of the parent model shapes its phase space
by creating a set of Casimir invariants as a direct consequence of the Jacobi
identity.
