L(q, 0, \xi) $. We hence recover the Clebsch optimal control problem discussed at the beginning of the section.

For the applications we have in mind, the optimal control constraint is a too strong condition. In the context of Yang–Mills theory, it amounts to requiring that the color-electromagnetic field vanishes and, for Einstein’s equation, it is equivalent to a zero extrinsic curvature.

Clebsch–Hamilton picture {#MomMapCon}
------------------------

In this section, we discuss the Hamiltonian counterpart of the Clebsch–Euler–Lagrange equations. Let $ E: \TBundle Q \times \LieA{g} \to \R $, $$E(q, \dot{q}, \xi) \defeq \dualPair*{\difpFrac{L}{\dot{q}}(q, \dot{q} + \xi \ldot q, \xi)}{\dot{q} + \xi \ldot q} - L(q, \dot{q} + \xi \ldot q, \xi),$$ be the energy function associated to the Clebsch–Lagrangian $ L $. The is defined by $$\label{eq:clebschLagrange:clebschLegendre}
    \mathrm{CL}: \TBundle Q \times \LieA{g} \to \CotBundle Q \times \LieA{g},
        \quad
        (q, \dot{q}, \xi) \mapsto \left(q, \difpFrac{L}{\dot{q}}(q, \dot{q} + \xi \ldot q, \xi), \xi \right).$$ We say that $ L $ is regular if $ \mathrm{CL} $ is a diffeomorphism. For a regular Clebsch–Lagrangian $ L $, the associated $ H: \CotBundle Q \times \LieA{g} \to \R $ is defined by $ H = E \circ \mathrm{CL}^{-1} $.

Clearly $ \mathrm{CL} $ is 