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

TL;DR

This paper addresses the challenge of incorporating inertial effects into Langevin equations by proposing a canonical formalism solution, revealing how the hierarchy of environments influences the Lagrangian structure.

Contribution

It introduces a canonical formalism to include inertial effects in Langevin equations, overcoming Ostrogradsky's theorem limitations, and analyzes the impact of environment hierarchy on the Lagrangian.

Findings

Canonical formalism resolves Ostrogradsky's instability in second order variational principles.

Hierarchy of environments significantly affects the Lagrangian structure.

Inertial effects can be incorporated consistently into stochastic dynamics.

Abstract

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