A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

TL;DR

This paper introduces a direct method for constructing standard and non-standard Lagrangians for dissipative one-dimensional systems with variable coefficients, expanding the theoretical framework and providing new formulations.

Contribution

It presents a novel direct approach to derive Lagrangians for dissipative systems with polynomial velocity dependence, including new formulations and generalizations of existing results.

Findings

Derived new Lagrangian formulations for dissipative systems

Found infinite families of Lagrangians for certain frictional systems

Generalized recent results on Lagrangian construction

Abstract

We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.