Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Autonomous second-order ODEs: a geometric approach

Antonio J. Pan-Collantes, Jose A. Alvarez-Garcia

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.

Autonomous second-order ODEs: a geometric approach

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.

Paper Structure

This paper contains 10 sections, 4 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jet bundles and second-order ODEs
  4. Riemannian geometry
  5. Riemannian metric associated to autonomous second-order ODEs
  6. Energy foliation
  7. Lagrangian mechanical systems
  8. Lagrangian for a particle in a gravitational field
  9. The damped harmonic oscillator
  10. Concluding remarks

Key Result

Proposition 3.1

Suppose equation ODE1vez satisfies $(\frac{\phi}{u_1})_{u_1}\neq 0$. If a smooth function $f$ is such that the curve $\hbox{j}^1 f$ is a geodesic, then $f$ is a solution of ODE1vez.

Theorems & Definitions (19)

  • Definition 3.1
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Example 3.1
  • Example 3.2
  • Definition 4.1
  • Remark 4.1
  • Remark 4.2
  • Example 4.1
  • ...and 9 more