  In this work, we study the reduction by a Lie group of symmetries of
variational collision avoidance probelms of multiple agents evolving on a
Riemannian manifold and derive necessary conditions for the reduced 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.
