In this work, we consider a mechanical system whose mass tensor implements a scalar product in a Riemannian manifold. This system is controlled with the help of forces and torques. A cost functional is minimized to achieve an optimal trajectory. In this contribution, this cost function is supposed to be an arbitrary regular function invariant under a change of coordinates. Optimal control evolution based on Pontryagin's principle induces a covariant second-order ordinary differential equation for an adjoint variable featuring the Riemann curvature tensor. This second order time evolution is derived in this contribution.