  The Einstein-Vlasov equations govern Einstein spacetimes filled with matter
which interacts only via gravitation. The matter, described by a distribution
function on phase space, evolves under the collisionless Boltzmann equation,
corresponding to the free geodesic motion of the particles, while the source of
the gravitational field is given by the stress-energy tensor defined in terms
of momenta of the distribution function. As no variational derivation of the
Einstein-Vlasov system appears to exist in the literature, we here set out to
fill this gap. In our approach we treat the matter as a generalized type of
fluid, flowing in the tangent bundle instead of the spacetime. We present the
actions for the Einstein-Vlasov system in both the Lagrangian and Eulerian
pictures.
