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Onsager and Machlup proposed a second order variational-principle in order toinclude inertial effects into the Langevin-equation, giving a Lagrangian withsecond order derivatives in time. This but violates Ostrogradysky's theorem,which proves that Lagrangians with higher than first order derivatives aremeaningless. As a consequence, inertial effects cannot be included in astandard way. By using the canonical formalism, we suggest a solution to thisfundamental problem. Furthermore, we provide elementary arguments about thehierarchy of immersions and actions between an ideal system and severalenvironments and show, that the structure of the Lagrangian sensitively dependson this hierarchy.

On Newton's equation of motion with friction and stochastic noise, the  Ostrogradsky-instability and the hierarchy of environments, An application of  the Onsager-Machlup theory II