Inverse variational problem for non-standard Lagrangians

TL;DR

This paper develops a systematic method to derive non-standard Lagrangians for dissipative systems by solving the inverse variational problem, providing new representations and applying to various models.

Contribution

It introduces a general approach to obtain NSLs from the first integral, improving upon ad hoc methods and enabling applications to diverse physical systems.

Findings

Derived new Lagrangian forms for dissipative equations

Applied method to Emden-type and Lotka-Volterra models

Showed the approach's general applicability

Abstract

The non-standard Lagrangians (NSLs) for dissipative-like dynamical systems were introduced in an ad hoc fashion rather than being derived from the solution of the inverse problem of variational calculus. We begin with the first integral of the equation of motion and solve the associated inverse problem to obtain some of the existing results for NSLs. In addition, we provide a number of alternative Lagrangian representations. The case studies envisaged by us include (i) the usual modified Emden-type equation, (ii) Emden-type equation with dissipative term quadratic in velocity and (iii) Lokta-Volterra model. We point out that our method is quite general for applications to other physical systems.