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

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 utility of the results, including an application to relativistic pp-waves.Key WordsDynamics of classical particles, autonomous and non-autonomous systems, second order differential equation on a Riemannian manifold, completeness of inextensible trajectories.MSC2010. Primary: 34A26, 34C40. Secondary: 37C60, 53D25, 70G45, 83C75.

Completeness of general pp -wave spacetimes and their impulsive limit

“…Similarly the global behaviour of geodesics in NPWs is governed by the behaviour of H at spatial infinity. The respective results will be discussed in the next section together with the stronger and newer results of [CRS12,CRS13].…”

“…Hence in the case of NPWs, which amounts to setting F = 0 and X = −∇ x V = ∇ x H, one obtains different types of results based on either of these conditions, see [CRS12,CRS13]. Explicitly we have Corollary 3.4 (Completeness of NPWs and classical pp-waves).…”

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

“…Remark 3.2. The inequality (18) mentions F ♯ t in order to clarify that the inclusion of force field components of 'electromagnetic' or 'Coriolis' type can only enlarge the domain of the maximal solutions (see also [17]). Actually, the lefthand side of Eq.…”

The globally hyperbolic metric splitting for non-smooth wave-type space-times

“…Candela, Romero, and Sánchez give sufficient conditions on a ensuring completeness of the trajectories of (2.5) in [8] (especially Theorem 2 and Subsection 3.2).…”

“…Then we are able to apply Theorem 2.3 for every ǫ and deduce that S k λ is a Cauchy hypersurface for every l ǫ (ǫ > 0 small) for any k ∈ R. Consequently, uniform bounds on ∂aǫ ∂u with respect to ǫ will be sufficient (cf. [8,Prop. 2]) for the completeness of these trajectories.…”