Jets, Lifts and Dynamics

We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie-Poisson reductions and with the techniques of present framework.