Autonomous second-order ODEs: a geometric approach

TL;DR

This paper introduces a geometric framework for autonomous second-order ODEs by defining a Riemannian metric on jet bundles, linking solutions to geodesics, and applying the approach to Lagrangian mechanics.

Contribution

It establishes a geometric approach to autonomous second-order ODEs using Riemannian metrics and introduces the concept of energy foliation, connecting differential equations with geometric structures.

Findings

Solutions correspond to geodesics of the defined metric

Energy foliation relates to classical energy concepts

Application to Lagrangian systems, including damped harmonic oscillator

Abstract

Given an autonomous second-order ordinary differential equation (ODE), we define a Riemannian metric on an open subset of the first-order jet bundle. A relationship is established between the solutions of the ODE and the geodesic curves with respect to the defined metric. We introduce the notion of energy foliation for autonomous ODEs, and highlight its connection to the classical energy concept. Additionally, we explore the geometry of the leaves of the foliation. Finally, the results are applied to the analysis of Lagrangian mechanical systems. In particular, we provide an autonomous Lagrangian for the damped harmonic oscillator.