Singular trajectory in the problem of simple pursuit on a manifold

Games with at most two minimal geodesics connecting two points in their phase space (manifold) are considered. It is shown that singular trajectories - envelopes of geodesics - may develop in such games. A necessary condition for the existence of singular motions (non-emptiness of the set (3.5)) is derived as a certain requirement from the geometrical properties of the phase manifold of the game. An algorithm to construct the singular motions is proposed and the Hamiltonian equations describing these motions are given. The sufficiency of the proposed construction is investigated numerically for particular examples. The paper generalizes and extends the previous study /1/.