  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.
