pairing $ \dualPairDot: \CotBundle Q \times \TBundle Q \to \R $. In line with our notation for points of $ \TBundle Q $, we will denote points in the cotangent bundle $ \CotBundle Q $ by pairs $ (q, p) $ with $ q \in Q $ and $ p \in \CotBundle_{q} Q $. The dual pairing yields an embedding of the cotangent bundle $ \CotBundle Q $ into the topological dual $ \TBundle' Q $ of the tangent bundle. In the following, we always assume that all occurring dual objects can be represented by points of the cotangent bundle $ \CotBundle Q $. In particular, the derivatives $ \difpFrac{L}{q} $ and $ \difpFrac{L}{\dot{q}} $ are viewed as maps $ \TBundle Q \to \CotBundle Q $ (and not merely to $ \TBundle' Q $). Regularity of $ L $ then means that $ \difpFrac{L}{\dot{q}} $ is a diffeomorphism between $ \TBundle Q $ and $ \CotBundle Q $. Similarly as for the cotangent bundle, we choose a non-degenerate pairing $ \kappa: \LieA{g}^* \times \LieA{g} \to \R $. This gives an embedding of $ \LieA{g}^* $ into the topological dual $ \LieA{g}' $. We silently assume that all occurring dual objects 