Singular Lagrangian and constrained Hamiltonian systems, generalized canonical formalism

Abstract

The relation between singular Lagrangian systems and the corresponding constrained Hamiltonian systems is studied in detail by making use of the « generalized canonical quantities » (g.c.q.). They are functions ofq, q andp and are independent ofq in the subspace in which the velocity-momentum relation (p = ∂L/∂q) holds. They are intimately related to the Lagrangian variables, on the one side, and have the canonical character, on the other. The generators of the Lagrangian transformations are constructed as the g.c.q. As in the standard case, they are conserved quantities when the Lagrangian is invariant under the transformations. The Hamiltonian is also a g.c.q. and is the generator of time translation. The canonical constraints are constructed from the singular Lagrangian systematically in the form of g.c.q. They are the generators of the transformations under which the Lagrangian is varied by quantities equal to the Lagrangian constraints.