Langevin memory kernel and noise from Lagrangian dynamics

The motion of a particle in a medium is studied using a Lagrangian description of the system with a realistic non-linear coupling between the particle and the medium. The force exerted on the particle by the medium is shown to consist of a damping force and noise. Under certain conditions the particle obeys a generalized Langevin equation. A systematic procedure for calculating the memory kernel and noise correlations in terms of the parameters and the noise obey a fluctuation dissipation theorem, leading to thermalization of the particle. The friction constant vanishes at T = 0.