Small oscillations of non-dissipative Lagrangian systems

TL;DR

This paper analyzes small oscillations in non-dissipative Lagrangian systems, providing solutions via quadratures and demonstrating that the general motion is a superposition of harmonic oscillations.

Contribution

It extends the analysis of small oscillations to non-dissipative systems with a new complexification method for solving linearized equations.

Findings

Solutions are obtained by quadratures.

The general integral is a superposition of harmonic oscillations.

A complexification of the solving algorithm is introduced.

Abstract

The small oscillations of an arbitrary scleronomous system subject to time-independent non dissipative forces are discussed. The linearized equations of motion are solved by quadratures. As in the conservative case, the general integral is shown to consist of a superposition of harmonic oscillations. A complexification of the resolving algorithm is presented.