Lagrangian descriptions of dissipative systems: a review

In this paper, we review classical and recent results on the Lagrangian description of dissipative systems. After having recalled Rayleigh extension of Lagrangian formalism to equations of motion with dissipative forces, we describe Helmholtz conditions, which represent necessary and sufficient conditions for the existence of a Lagrangian function for a system of differential equations. These conditions are presented in different formalisms, some of them published in the last decades. In particular, we state the necessary and sufficient conditions in terms of multiplier factors, discussing the conditions for the existence of equivalent Lagrangians for the same system of differential equations. Some examples are discussed, to show the application of the techniques described in the theorems stated in this paper.

Automated summary

This paper reviews classical and recent results on the Lagrangian description of dissipative systems, focusing on the Helmholtz conditions for the existence of a Lagrangian function for a system of differential equations. The necessary and sufficient conditions are discussed in terms of multiplier factors, exploring the existence of equivalent Lagrangians for the same system. Examples are provided to illustrate the application of the techniques described in the theorems presented in the paper.