a diffeomorphism if and only if the fiber derivative $$\TBundle Q \to \CotBundle Q,
        \quad 
        (q, v) \mapsto \left(q, \difpFrac{L}{\dot{q}}(q, v, \xi)\right)$$ is a diffeomorphism for every $ \xi \in \LieA{g} $. Moreover, the Clebsch–Hamiltonian $ H $ coincides with the Hamiltonian corresponding to the Clebsch–Lagrangian $ L $ via the ordinary Legendre transformation with $ \xi $ viewed as a parameter, that is, $$\label{eq:hamiltonian}
        H \left(q, p , \xi \right) = \dualPair*{p}{v} - L(q, v, \xi)\, ,$$ where the relation $ p = \difpFrac{L}{\dot{q}}(q, v, \xi) $ defines $ v $ as a function of $ q, p $ and $ \xi $.

Let $ K $ be a torsion-free connection in $ \TBundle Q $ and let $ \bar{K} $ be the dual connection in $ \CotBundle Q $. By , $ \bar{K} $ induces a decomposition $$\label{eq:clebschLagrange:hamilton:decompCotBundle}
    \TBundle (\CotBundle Q) \isomorph \CotBundle Q \times_Q \TBundle Q \times_Q \CotBundle Q$$ and we will write points in $ \TBundle (\CotBundle Q) $ as tuples $ (q, p, \delta q, \delta p) $ with $ p, \delta p \in \CotBundle_q Q $ and $ \delta q \in \TBundle_q Q $. Note that the component $ \delta p $ depends on 