The inverse problem for Lagrangian systems with certain non-conservative forces

TL;DR

This paper explores generalized inverse problems for Lagrangian systems, providing coordinate-free conditions for representing systems with non-conservative forces, including Rayleigh dissipation and gyroscopic forces.

Contribution

It introduces new coordinate-free criteria for expressing second-order systems as Lagrangian with non-conservative forces, extending classical inverse problem results.

Findings

Derived coordinate-free conditions for non-conservative Lagrangian systems

Extended inverse problem to include Rayleigh dissipation forces

Provided criteria for systems with gyroscopic forces

Abstract

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.