the anonymous reviewer for his/her comments and suggestions on an earlier draft, which significantly contributed to improving the quality of the paper. We gratefully acknowledge support of the Max Planck Institute for Mathematics in the Sciences in Leipzig and of the University of Leipzig.

Clebsch–Lagrange variational principle {#sec:clebschLagrange}
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In [@Gay-BalmazRatiu2011], @Gay-BalmazRatiu2011 study a class of optimal control problems associated to group actions. The special feature of the is that the control variables are Lie algebra valued and couple to the state variables via the symmetry group action. Let $ G $ be a Lie group that acts smoothly on the smooth manifold $ Q $ and let $ \LieA{g} $ be its Lie algebra. Given a smooth cost function $ l: Q \times \LieA{g} \to \R $, the Clebsch optimal control problem consists in finding curves $ t \mapsto q(t) $ and $ t \mapsto \xi(t) $ such that $$s[q, \xi] = \int_0^T l\bigl(q(t), \xi(t)\bigr) \dif t$$ is minimized subject to the optimal control constraint $ \dot q(t) = - \xi(t) \ldot q(t) $ and the endpoint constraints $ q(0) = q_i $ and $ q(T) = q_f $. Here, $ \xi \ldot q $ is the fundamental vector 