contrast to the fiber derivative $ \difpFrac{L}{\dot{q}} $, the partial derivative $ \difpFrac{L}{q} $ depends on the choice of a connection in $ \TBundle Q $. In summary, we express $ \tangent L: \TBundle (\TBundle Q) \to \R $ as $$\label{eq:clebschLagrange:tangentLagrange}
    \tangent L(q, \dot{q}, \diF q, \diF \dot{q}) = \dualPair*{\difpFrac{L}{q}(q, \dot{q})}{\diF q} + \dualPair*{\difpFrac{L}{\dot{q}}(q, \dot{q})}{\diF \dot{q}}.$$ This relation is the intrinsic geometric version of the corresponding coordinate expression found in the physics literature. For $ L: \TBundle Q \times \LieA{g} \to \R $, the derivative of $ L $ at $ (q, \dot q, \xi) \in \TBundle Q \times \LieA{g} $ in the $ \LieA{g} $-direction yields an element of the topological dual $ \LieA{g}' $ and will be denoted by $ \difpFrac{L}{\xi}(q, \dot q, \xi) $.

In infinite dimensions, one needs to be careful with the notion of a cotangent bundle, because the canonical candidate — the fiberwise topological dual $ \TBundle' Q $ of the tangent bundle — fails to be a *smooth* bundle. Following [@DiezRudolphReduction], we define the cotangent bundle $ \CotBundle Q $ to be some smooth vector bundle[^1] over $ Q $ which is fiberwise in duality with the tangent bundle relative to some chosen 