Alternative Lagrangians obtained by scalar deformations

TL;DR

This paper investigates conditions under which non-conservative second-order differential equations can be derived from a deformed Lagrangian, expanding the class of systems with explicit Lagrangian formulations.

Contribution

It establishes necessary and sufficient conditions for scalar deformations of Lagrangians that yield equivalent equations, including non-conservative systems.

Findings

Derived conditions for scalar Lagrangian deformations

Provided examples of deformed Lagrangians for non-conservative systems

Extended the class of systems with explicit Lagrangian descriptions

Abstract

We study non-conservative like SODEs admitting explicit Lagrangian descriptions. Such systems are equivalent to the system of Lagrange equations of some Lagrangian $L$, including a covariant force field which represents non-conservative forces. We find necessary and sufficient conditions for the existence of a differentiable function $\Phi:\mathbb{R}\rightarrow\mathbb{R}$ such that the initial system is equivalent to the system of Euler-Lagrange equations of the deformed Lagrangian $\Phi(L)$. We give various examples of such deformations.