Completeness of first and second order ODE flows and of Euler-Lagrange equations

TL;DR

This paper investigates the conditions under which solutions to first and second order ODEs on complete Riemannian manifolds are complete, with applications to Lagrangian mechanics and gravitational waves.

Contribution

It provides new results on the completeness of solutions to ODEs and Euler-Lagrange equations on manifolds with time-dependent metrics.

Findings

Established criteria for solution completeness in Riemannian manifolds.

Applied results to Lagrangian mechanics and gravitational wave models.

Abstract

Two results on the completeness of maximal solutions to first and second order ordinary differential equations (or inclusions) over complete Riemannian manifolds, with possibly time-dependent metrics, are obtained. Applications to Lagrangian mechanics and gravitational waves are given.