H $ in the $ \xi $-direction is given by $$\begin{split}
        \difpFrac{H}{\xi}(q, p, \xi)
            &= \dualPair*{p}{\difpFrac{v}{\xi}} - \dualPair*{\difpFrac{L}{\dot q}(q, v, \xi)}{\difpFrac{v}{\xi}} - \difpFrac{L}{\xi}(q, v, \xi)
            \\ 
            &= - \difpFrac{L}{\xi}(q, \dot q + \xi \ldot q, \xi).
    \end{split}$$ Hence, the constraint  is equivalent to .

In the proof of \[prop:clebschLagrange:legendreDirect\], we have seen that the first equation in  is equivalent to

(q, p, ) = - p + .

The constraint equation  has a natural reformulation in terms of the momentum map of the lifted $ G $-action on the cotangent bundle $ \CotBundle Q $. As usual, $ \CotBundle Q $ carries a canonical $ 1 $-form $ \theta $ defined by $$\label{CanForm}
    \theta_{(q,p)} (X) = \dualPair{p}{\tangent_{(q,p)} \pi (X)}_q,$$ where $ X \in \TBundle_{(q,p)} (\CotBundle Q) $ and $ \pi $ is the natural projection of $ \CotBundle Q $. Correspondingly, the canonical symplectic form is $ \omega = \dif \theta $. Recall the $ G $-action on $ \TBundle Q $ written as $ g \ldot \dot{q} \in \TBundle_{g \cdot q} Q $ for $ g \in G $ and $ \dot{q} \in \TBundle_q Q $. We assume that the relation $$\dualPair{g \cdot p}{\dot{q}}
        = \dualPair{p}{g^{-1} \ldot \dot{q}}$$ for 