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We straight-forwardly derive the Onsager-Machlup Lagrangian from theFokker-Planck equation and show that friction and dissipation are a naturalproperty of the equation of motion. We develop a method to calculate the localvariance $\sigma_{2}\,b(q)^{2}$ and identify this function as aHelmholtz-factor. In both meanings the function $b(q)$ describes properties ofthe environment. For application, we examine the free fall through a barometricmedium and model a blow of wind by a solitonic pulse running through themedium. We treat harmonic oscillators immersed in a thermal bath, findingintuitive as well as counter-intuitive phenomena of friction. By allowing thetemperature to be time-dependent, the dynamical process of cooling and heatingbecomes self-consistently available. We find a state of dynamical balancebetween system and environment. Last, we show that dynamical balance is relatedto adiabatic thermodynamic processes. In a special case, dynamical balance caninduce a real phase-transition.

Lagrangian dynamics in inhomogeneous and thermal environments, An application of the Onsager-Machlup theory I