This phase space is essentially T *T Q, 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 (qa,qa,p ,p) a a. The usual Lagrangian (or velocity) phase space TQ and the Hamiltonian (or momentum) phase space T*Q are special submanifolds of T*T Q.