This phase space is essentially ∗ T TQ, the cotangent bundle of the tangent bundle of the configuration manifold Q, endowed with the natural symplectic structure, and can be interpreted as the collection of quadruples a a (q , ˙q ,pa,p˙a). The usual Lagrangian (or velocity) phase space T Q and the Hamiltonian (or momentum) phase space ∗ T Q are special submanifolds of ∗ T T Q.