in $ Q $ with given endpoints $ q_i $ and $ q_f $ is a smooth infinite-dimensional manifold, [@Wockel2014 Theorem 7.6]. Similarly, as $ \LieA{g} $ is an affine space, the space $ \sFunctionSpace(I, \LieA{g}) $ of paths in $ \LieA{g} $ is a smooth manifold, too. Since the Lagrangian is smooth, the action $ S $ is a smooth function on $ \sFunctionSpace_{q_i q_f}(I, Q) \times \sFunctionSpace(I, \LieA{g}) $. The variational principle amounts to looking for extrema of $ S $.

Let $ \bigl(q_\varepsilon(t), \dot{q}_\varepsilon(t)\bigr) $ be a smooth perturbation of the curve $ \bigl(q(t), \dot{q}(t)\bigr) $ in $ \TBundle Q $. In line with usual notation, we write $$\label{eq:clebschLagrange:clebschEulerLagrangeDirect:variation}
         \difFracAt{}{\varepsilon}{0}\bigl(q_\varepsilon(t), \dot{q}_\varepsilon(t)\bigr) \equiv \bigl(\diF q(t), \diF \dot{q}(t)\bigr) \in \TBundle_{q(t)} Q \times \TBundle_{q(t)} Q$$ for the corresponding curve in $ \TBundle(\TBundle Q) $ under the isomorphism . The curve $ t \mapsto \bigl(\diF q(t), \diF \dot{q}(t)\bigr) $ should be considered as a tangent vector field along the curve $ t \mapsto q(t) $, which may be viewed as an tangent vector to $ \sFunctionSpace_{q_i q_f}(I, Q) $ at that curve. Let $ \phi: I \times \R \to Q $ be defined by $ \phi(t, \varepsilon) = q_\varepsilon(t) $, and 