Lie Algebra of Hamiltonian Vector Fields and the Poisson-Vlasov Equations

TL;DR

This paper explores the geometric structures underlying the Poisson-Vlasov equations, revealing a Lie algebra framework for Hamiltonian vector fields, and deriving the equations as momentum maps within this geometric setting.

Contribution

It introduces a Lie algebraic perspective on Hamiltonian vector fields and their relation to the Poisson-Vlasov equations, including new geometric decompositions and lifts.

Findings

Decomposition of vector fields into Hamiltonian and complementary parts.

Representation of momentum-Vlasov equations via cotangent lifts.

Identification of gauge symmetries leading to the Poisson equation.

Abstract

We introduce natural differential geometric structures underlying the Poisson-Vlasov equations in momentum variables. We decompose the space of all vector fields over particle phase space into a semi-direct product algebra of Hamiltonian vector fields and its complement. The latter is related to dual space of Lie algebra. Lie algebra of Hamiltonian vector fields is isomorphic to the space of all Lagrangian submanifolds with respect to Tulczyjew symplectic structure. This is obtained as tangent space at the identity of the group of canonical diffeomorphisms represented as space of sections of a trivial bundle. We obtain the momentum-Vlasov equations as vertical equivalence of complete cotangent lift of Hamiltonian vector field generating particle motion. Vertical representatives can be described by holonomic lift from a Whitney product to a Tulczyjew symplectic space. A generalization of complete cotangent lift is obtained by a Lie algebra homomorphism from the algebra of symmetric contravariant tensor fields with Schouten concomitant to the Lie algebra of Hamiltonian vector fields. Momentum maps for particular subalgebras result in plasma-to-fluid map in momentum variables. We exhibit dynamical relations between Lie algebras of Hamiltonian vector fields and of contact vector fields, in particular; infinitesimal quantomorphisms. Gauge symmetries of particle motion are extended to tensorial objects including complete lift of particle motion. Poisson equation is then obtained as zero value of momentum map for the Hamiltonian action of gauge symmetries for kinematical description.