Lagrangian Dynamics in Non-Flat Universes and Non-Linear Gravitational Evolution

We present a new formalism which allows to derive the general Lagrangian dynamical equations for the motion of gravitating particles in a non--flat Friedmann universe with arbitrary density parameter $\Omega$ and no cosmological constant. We treat the set of particles as a Newtonian collisionless fluid. The non--linear dynamical evolution of the fluid element trajectories is then described up to the third--order expansion in Lagrangian coordinates. This work generalizes recent investigations carried out by Bouchet \etal (1992) and Buchert (1994).