Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields

Abstract

We introduce natural differential geometric structures underlying thePoisson-Vlasov equations in momentum variables. First, we decompose thespace of all vector fields over particle phase space into a semi-directproduct algebra of Hamiltonian vector fields and its complement. The latteris related to dual space of the Lie algebra. We identify generators ofhomotheties as dynamically irrelevant vector fields in the complement. Liealgebra of Hamiltonian vector fields is isomorphic to the space of allLagrangian submanifolds with respect to Tulczyjew symplectic structure. Thisis obtained as tangent space at the identity of the group of canonicaldiffeomorphisms represented as space of sections of a trivial bundle. Weobtain the momentum-Vlasov equations as vertical equivalence, orrepresentative, of complete cotangent lift of Hamiltonian vector fieldgenerating particle motion. Vertical representatives can be described byholonomic lift from a Whitney product to a Tulczyjew symplectic space. Weshow that vertical representatives of complete cotangent lifts form anintegrable subbundle of this Tulczyjew space. We exhibit dynamical relationsbetween Lie algebras of Hamiltonian vector fields and of contact vectorfields, in particular; infinitesimal quantomorphisms on quantization bundle.Gauge symmetries of particle motion are extended to tensorial objectsincluding complete lift of particle motion. Poisson equation is thenobtained as zero value of momentum map for the Hamiltonian action of gaugesymmetries for kinematical description.