  We consider the physical model of a classical mechanical system (called
"small system") undergoing repeated interactions with a chain of identical
small pieces (called "environment"). This physical setup constitutes an
advantageous way of implementing dissipation for classical systems, it is at
the same time Hamiltonian and Markovian. This kind of model has already been
studied in the context of quantum mechanical systems, where it was shown to
give rise to quantum Langevin equations in the limit of continuous time
interactions ([2]), but it has never been considered for classical mechanical
systems yet. The aim of this article is to compute the continuous limit of
repeated interactions for classical systems and to prove that they give rise to
particular stochastic differential equations in the limit. In particular we
recover the usual Langevin equations associated to the action of heat baths. In
order to obtain these results, we consider the discrete-time dynamical system
induced by Hamilton's equations and the repeated interactions. We embed it into
a continuous-time dynamical system and we compute the limit when the time step
goes to 0. This way we obtain a discrete-time approximation of stochastic
differential equation, considered as a deterministic dynamical system on the
Wiener space, which is not exactly of the usual Euler scheme type. We prove the
Lp and almost sure convergence of this scheme. We end up with applications to
concrete physical exemples such as a charged particule in an uniform electric
field or a harmonic interaction. We obtain the usual Langevin equation for the
action of a heat bath when considering a damped harmonic oscillator as the
small system.
