Back to Maupertuis' least action principle for dissipative systems: not all motions in Nature are most energy economical

TL;DR

This paper demonstrates that Maupertuis' least action principle can be extended to dissipative systems, revealing that not all motions are energy-efficient and establishing a connection between least action and least dissipation in certain cases.

Contribution

It introduces a variational approach for damped motions using Maupertuis' principle, linking energy dissipation optimization to classical least action.

Findings

Least action principle applies to certain dissipative motions.

Not all classical motions follow the path of least energy dissipation.

Least action is equivalent to least dissipation for stationary and Stokes drag motions.

Abstract

It is shown that an oldest form of variational calculus of mechanics, the Maupertuis least action principle, can be used as a simple and powerful approach for the formulation of the variational principle for damped motions, allowing a simple derivation of the Lagrangian mechanics for any dissipative systems and an a connection of the optimization of energy dissipation to the least action principles. On this basis, it is shown that not all motions of classical mechanics obey the rule of least energy dissipation or follow the path of least resistance, and that the least action is equivalent to least dissipation for two kinds of motions : all stationary motions with constant velocity and all motions damped by Stokes drag.