  In this work, we consider optimal control problems for mechanical systems on
vector spaces with fixed initial and free final state and a quadratic Lagrange
term. Specifically, the dynamics is described by a second order ODE containing
an affine control term and we allow linear coordinate changes in the
configuration space. Classically, Pontryagin's maximum principle gives
necessary optimality conditions for the optimal control problem. For smooth
problems, alternatively, a variational approach based on an augmented objective
can be followed. Here, we propose a new Lagrangian approach leading to
equivalent necessary optimality conditions in the form of Euler-Lagrange
equations. Thus, the differential geometric structure (similar to classical
Lagrangian dynamics) can be exploited in the framework of optimal control
problems. In particular, the formulation enables the symplectic discretisation
of the optimal control problem via variational integrators in a straightforward
way.
