Analytic representation of discrete and continuous mechanical systems

TL;DR

This paper demonstrates how self-adjoint differential equations can be used to derive explicit Lagrangian and analytic representations for both discrete and continuous mechanical systems, including nonlinear and damped oscillators.

Contribution

It introduces a unified method using self-adjoint forms to find explicit time-dependent and independent representations of mechanical systems, extending to nonlinear evolution equations.

Findings

Explicit time-dependent Lagrangian for discrete systems

Analytic representation of damped harmonic oscillator revealing Bateman's equation

Application to nonlinear evolution equations

Abstract

We investigate how the theory of self-adjoint differential equations alone can be used to provide a satisfactory solution of the inverse vatiational problem. For the discrete system, the self-adjoint form of the Newtonian equation allows one to find an explicitly time-dependent Lagrangian representation. On the other hand, the same Newtonian equation in conjunction with its adjoint forms a natural basis to construct an explicitly time-independent analytic representation of the system. This approach when applied to the equation of damped harmonic oscillator help one disclose the mathematical origin of the Bateman image equation. We have made use of a continuum analog of the same approach to find the Lagrangian or analytic representation of nonlinear evolution equations.