Lagrangian dynamics in non-flat universes and non-linear gravitational evolution

We present a new formalism that allows the derivation of the general Lagrangian dynamical equations of motion for gravitating particles in a non-flat Friedmann universe with arbitrary density parameter Ω 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 et al. and Buchert.