On the completeness of trajectories for some mechanical systems

TL;DR

This paper revisits classical methods for ensuring the completeness of trajectories in mechanical systems, exploring extensions to advanced geometric frameworks and discussing open questions and future directions.

Contribution

It provides a thorough analysis of classical completeness criteria and proposes potential extensions to infinite-dimensional, Finsler, and pseudo-Riemannian settings.

Findings

Revisited classical completeness criteria for mechanical systems.

Explored extensions to infinite-dimensional manifolds and advanced geometries.

Discussed open questions and future research directions.

Abstract

The classical tools which ensure the completeness of vector fields and second order differential equations for mechanical systems are revisited. Possible extensions in three directions are discussed: infinite dimensional Banach and Hilbert manifolds, Finsler metrics and pseudo-Riemannian spaces, including links with some relativistic spacetimes. Special emphasis is taken in the cleaning up of known techniques, the statement of open questions and the exploration of prospective frameworks.