Completeness of the Trajectories of Particles Coupled to a General Force Field

TL;DR

This paper investigates the conditions under which solutions to a second order differential equation on a Riemannian manifold, influenced by general forces, can be extended, generalizing classical Lagrangian and Hamiltonian results with practical examples.

Contribution

It extends classical results on solution extendability to more general forces, including time-dependent and velocity-dependent cases, on Riemannian manifolds.

Findings

Results include conditions for solution extendability in general force fields.

Refinements are provided for autonomous and potential-derived forces.

Applications demonstrate the optimality and relevance of the theoretical results.

Abstract

We analyze the extendability of the solutions to a certain second order differential equation on a Riemannian manifold $(M,g)$, which is defined by a general class of forces (both prescribed on $M$ or depending on the velocity). The results include the general time-dependent anholonomic case, and further refinements for autonomous systems or forces derived from a potential are obtained. These extend classical results for Lagrangian and Hamiltonian systems. Several examples show the optimality of the assumptions as well as the applicability of the results, including an application to relativistic pp-waves.