Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments

TL;DR

This paper explores Hamiltonian closures of the 1D Vlasov equation via phase-space moments, revealing how such closures preserve Hamiltonian structure and introduce Casimir invariants.

Contribution

It provides conditions for Hamiltonian closures of the Vlasov equation and fully solves specific examples, highlighting the role of the Jacobi identity.

Findings

Hamiltonian closures impose specific conditions via the Jacobi identity.

Certain families of closures are completely characterized.

Casimir invariants naturally arise from Hamiltonian constraints.

Abstract

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.