In this talk, we derive the necessary conditions for optimality in a second-order variational problem on Riemannian homogeneous spaces by making use of the second fundamental form, reinterpreted as a connection on a horizontal distribution on the underlying Lie group. We assume that the action functional admits partial symmetry over some parameter manifold, and use this to reduce the necessary conditions by symmetry. We then study the design of artificial potentials for the obstacle avoidance and discuss applications to path-planning of robotic systems.