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JGM

We introduce natural differential geometric structures underlying the
Poisson-Vlasov equations in momentum variables. First, 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 the Lie algebra. We identify generators of
homotheties as dynamically irrelevant vector fields in the complement. 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, or
representative, 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. We
show that vertical representatives of complete cotangent lifts form an
integrable subbundle of this Tulczyjew space. We exhibit dynamical relations
between Lie algebras of Hamiltonian vector fields and of contact vector
fields, in particular; infinitesimal quantomorphisms on quantization bundle.
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.

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