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 $\mathrm{\Omega }$ 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).