the choice of the connection $ \bar{K} $. Given a smooth function $ H: \CotBundle Q \times \LieA{g} \to \R $ (not necessarily obtained via a Clebsch–Legendre transformation), we decompose the derivative $ \tangent H: \TBundle(\CotBundle Q) \to \R $ relative to  as follows: $$\label{eq:clebschLagrange:hamilton:decompTHamilton}
    \tangent H (q, p, \delta q, \delta p) 
        = \dualPair*{\difpFrac{H}{q}(q, p)}{\delta q} + \dualPair*{\delta p}{\difpFrac{H}{p}(q, p)},$$ where we assume that the partial derivatives are maps $ \difpFrac{H}{q}: \CotBundle Q \to \CotBundle Q $ and $ \difpFrac{H}{p}: \CotBundle Q \to \TBundle Q $. We emphasize that, according to , $ \difpFrac{H}{p} $ is the intrinsically defined fiber derivative of $ H $ while $ \difpFrac{H}{q} $ depends on the choice of the connection $ \bar{K} $.

\[prop:clebschLagrange:legendreDirect\] For a regular Clebsch–Lagrangian $ L: \TBundle Q \times \LieA{g} \to \R $ with associated Clebsch–Hamiltonian $ H: \CotBundle Q \times \LieA{g} \to \R $, the curves $ t \mapsto q(t) \in Q $ and $ t \mapsto \xi(t) \in \LieA{g} $ are solutions of the Clebsch–Lagrange variational problem  if and only if the curves $$t \mapsto (q(t), p(t), \xi(t)) = \mathrm{CL}\bigl(q(t), \dot{q}(t), \xi(t)\bigr)$$ satisfy the following equations

\[eq:clebschLagrange:hamiltonianDirect\]

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