a smooth vector bundle over $ N $. The connection on $ E $ yields via pull-back a connection in $ f^* E $ and thus induces a covariant derivative $$\nabla^f_Y: \sSectionSpace(f^* E) \to \sSectionSpace(f^* E),
    \quad \psi \mapsto K\bigl(\tangent \psi (Y)\bigr)$$ for every $ Y \in \VectorFieldSpace(N) $. In particular, for a curve $ \gamma: \R \supseteq I \to M $, we write $$\label{eq:connection:alongCurve}
    \DifFrac{}{t} \equiv \nabla^\gamma_{\difp_t}: \sSectionSpace(\gamma^* E) \to \sSectionSpace(\gamma^* E)$$ for the covariant derivative along $ \gamma $ in the direction of the canonical vector field $ \difp_t $ on $ I $.

Returning to our original setting, let $ Q $ be a smooth manifold and let $ L: \TBundle Q \to \R $ be a smooth Lagrangian. The $ \difFibre L: \TBundle Q \to \TBundle' Q $ of $ L $ is defined by $$\label{FibDer}
    \dualPair{\difFibre L (v)}{w} = \difFracAt{}{\varepsilon}{0} L(v + \varepsilon w)\, ,$$ where $ v, w \in \TBundle_q Q $, and $ \TBundle' Q $ denotes the fiberwise dual of the tangent bundle. In order to stay close to the usual physics notation, the fiber derivative $ \difFibre L: \TBundle Q \to \TBundle' Q $ will be written as $ \difpFrac{L}{\dot q} 