On the analytic representation of Newtonian systems

TL;DR

This paper demonstrates how self-adjoint differential equations can be used to find both direct and indirect Lagrangian representations of Newtonian systems, including nonlinear cases, through analytic and adjoint formulations.

Contribution

It introduces a novel approach using self-adjoint theory to derive explicit Lagrangian forms for Newtonian systems, including damped and nonlinear equations.

Findings

Derived time-dependent Lagrangian for damped harmonic oscillator

Constructed time-independent Lagrangian via adjoint equations

Extended method to nonlinear differential equations

Abstract

We show that the theory of self-adjoint differential equations can be used to provide a satisfactory solution of the inverse variational problem in classical mechanics. A Newtonian equation when transformed to the self-adjoint form allows one to find an appropriate Lagrangian representation (direct analytic representation) for it. On the other hand, the same Newtonian equation in conjunction with its adjoint provides a basis to construct a different Lagrangian representation (indirect analytic representation) for the system. We obtain the time-dependent Lagrangian of the damped Harmonic oscillator from the self-adjoint form of the equation of motion and at the same time identify the adjoint of the equation with the so called Bateman image equation with a view to construct a time-independent indirect Lagrangian representation. We provide a number of case studies to demonstrate the usefulness of the approach derived by us. We also present similar results for a number of nonlinear differential equations by using an integral representation of the Lagrangian function and make some useful comments.