Lagrangian Dynamics of Collisionless Matter

The non--linear dynamics of self--gravitating irrotational dust is analyzed in a general relativistic framework, using synchronous and comoving coordinates. Writing the equations in terms of the metric tensor of the spatial sections orthogonal to the fluid flow allows an unambiguous expansion in inverse powers of the speed of light. The Newtonian and post--Newtonian approximations are derived in Lagrangian form. A general formula for the gravitational waves generated by the non--linear evolution of cosmological perturbations is given. It is argued that a stochastic gravitational--wave background is produced by non--linear cosmic structures, with present--day closure density $\Omega_{gw} \sim 10^{-5}$ -- $10^{-6}$ on Mpc scale.