Stochastic embedding of dynamical systems

Most physical systems are modeled 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 behavior of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modeled 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. (c) 2007 American Institute of Physics.