  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).
