  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.
