  In this article we introduce a variational approach to collision avoidance of
multiple agents evolving on a Riemannian manifold and derive necessary
conditions for extremals. The problem consists of finding non-intersecting
trajectories of a given number of agents, among a set of admissible curves, to
reach a specified configuration, based on minimizing an energy functional that
depends on the velocity, covariant acceleration and an artificial potential
function used to prevent collision among the agents. The results are validated
through numerical experiments on the manifolds $\mathbb{R}^{2}$ and $S^2$.
