  Most physical systems are modelled by an ordinary or a partial differential
equation, like the n-body problem in celestial mechanics. In some cases, for
example when studying the long term behaviour of the solar system or for
complex systems, there exist elements which can influence the dynamics of the
system which are not well modelled or even known. One way to take these
problems into account consists of looking at the dynamics of the system on a
larger class of objects, that are eventually stochastic. In this paper, we
develop a theory for the stochastic embedding of ordinary differential
equations. We apply this method to Lagrangian systems. In this particular case,
we extend many results of classical mechanics namely, the least action
principle, the Euler-Lagrange equations, and Noether's theorem. We also obtain
a Hamiltonian formulation for our stochastic Lagrangian systems. Many
applications are discussed at the end of the paper.
