field at $ q \in Q $ generated by the action of $ \xi \in \LieA{g} $. The Pontryagin maximum principle shows that the control variable $ \xi $ satisfies the (generalized) Euler-Poincaré equation, see [@Gay-BalmazRatiu2011 Theorem 7.1]. This observation allows one to give an optimal control formulation for many systems including the heavy top and the compressible or magnetohydrodynamic fluid flow.

Clebsch–Lagrangian picture
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We now describe a novel variational principle which draws its inspiration from the Clebsch optimal control problem and which will turn out to be useful for relativistic field theories with constraints. The idea is to dismiss the optimal control constraint and to treat, instead, the quantity $ \dot q(t) + \xi(t) \ldot q(t) $ as an effective velocity in the Lagrangian formulation. As above, let $ Q $ be a smooth manifold and let $ \TBundle Q $ be its tangent bundle. Assume that a Lie group $ G $ acts smoothly on $ Q $. The configuration space $ Q $ as well as the Lie group $ G $ are assumed to be (infinite-dimensional) Fréchet manifolds. We refer to [@Neeb2006; @Hamilton1982] for the differential calculus in Fréchet spaces. In order to stay close 