to physics notation, we will usually denote points in $ \TBundle Q $ by pairs $ (q, \dot{q}) $, where $ q \in Q $ stands for the base point of the vector $ \dot{q} \in \TBundle_q Q $. Sometimes, we will also write a point in $ \TBundle Q $ as a pair $ (q, v) $ with $ q \in Q $ and $ v \in \TBundle_q Q $. Moreover, we use the dot notation $ g \cdot q $ for the action of $ g \in G $ on $ q \in Q $. For the derivative of the action a similar notation using lower dots is employed, , the fundamental vector field $ \xi_* $ generated by $ \xi \in \LieA{g} $ is written as $ \xi_* (q) = \xi \ldot q \in \TBundle_q Q $ and the lifted $ G $-action on $ \TBundle Q $ has the form $ g \ldot X_q \in \TBundle_{g \cdot q} Q $ for $ X_q \in \TBundle_q Q $.

Given a smooth function $ L: \TBundle Q \times \LieA{g} \to \R $, we consider the variational principle $ \diF S = 0 $ for curves $ t \mapsto 