$ g \in G, \dot{q} \in \TBundle_q Q $ and $ p \in \CotBundle_q Q $, defines a lift to $ \CotBundle Q $ of the $ G $-action on $ Q $ (which is automatic in finite dimensions). The lifted action on $ \CotBundle Q $ is symplectic, and the associated $ G $-equivariant momentum map $ J: \CotBundle Q \to \LieA{g}^* $ (if it exists) is defined by $$\label{eq:cotangentBundle:momentumMapDef}
    \kappa\left(J(q, p), \xi\right)
        = \dualPair{p}{\xi \ldot q}.$$ In infinite dimensions, the momentum map may not exist in pathological cases, [@DiezRudolphReduction Example 2.2], and we thus have to assume its existence in the forthcoming. The right-hand side of  is exactly what occurs in the constraint  and we hence arrive at the following reformulation of the Clebsch–Euler–Lagrange equations .

\[prop:clebschLagrange:legendre\] In the setting of \[prop:clebschLagrange:legendreDirect\], assume additionally that the lifted $ G $-action on the cotangent bundle $ \CotBundle Q $ has a momentum map $ J: \CotBundle Q \to \LieA{g}^* $ (which is automatic in finite dimensions). Then, the Clebsch–Euler–Lagrange equations are equivalent to the following set of equations:

\[eq:clebschLagrange:hamiltonian\]

\[eq:clebschLagrange:hamiltonian:hamiltonian\] (q, p, ) = - p - |[K]{}(p), (q, p, ) = q + q,\
\[eq:clebschLagrange:hamiltonian:constraint\] J(q, p) = 