  We study the reduction of degrees of freedom for the equations that determine
necessary optimality conditions for extrema in an optimal control problem for a
multiagent system by exploiting the physical symmetries of agents, where the
kinematics of each agent is given by a left-invariant control system. Reduced
optimality conditions are obtained using techniques from variational calculus
and Lagrangian mechanics. A Hamiltonian formalism is also studied, where the
problem is explored through an application of Pontryagin's maximum principle
for left-invariant systems, and the optimality conditions are obtained as
integral curves of a reduced Hamiltonian vector field. We apply the results to
an energy-minimum control problem for multiple unicycles.
