Poincare-Cartan Invariant Form and Dynamical Systems with Constraints

Abstract

A formulation of constrained dynamical systems is developes on the basis of the Poincaré-Cartan invariant form both in the Lagrangian and in the Hamiltonian formalism. The exterior differential form is used in analysis of the equations of motion. It is shown that by requiring the consistent correspondence of the Lagrangian formalism with the Hamiltonian one, the ambiguity in the choice of Hamiltonian in the Dirac theory can be removed and the total Hamiltonian, the generator of evolution of a constrained system, is uniquely determined, except for arbitrary gauge functions. The (n × n) singular Hessian matrix with the rank n − r, has eigenvectors ταi(α = 1 ∼ r) belonging to the zero eigenvalue. Those ταi are correlated with primary constraints Φα(q, p) in the phase space by ταi = ∂Φα/∂pi. In the vector field which generates evolution of the system, ταi appear accompanying undetermined coefficient functions, in proper response to Φα in the Hamiltonian in the Dirac theory. In the Lagrangian formalism, integrable equations which contain ταi with the undetermined coefficient functions are obtained. Some of the coefficient functions are determined by integrability conditions, but others remain arbitrary which give gauge freedom. If the rank of Aij is reduced further to n − r − r by taking account of (secondary) constraints χσ, extra eigenvectors τβi exist under mod(χσ). The τβi enter in the formulation on the same footing with ταi. Then the generalized Hamiltonian is given by Hg = H + vαΦα + vβΦβ where τβi = ∂Φβ/∂pi and H is the canonical Hamiltonian. Φα and Φβ are called “intrinsic constraints” in order to distinguish them from χσ. It is shown that first class intrinsic constraints are associated with gauge freedom, but those of χσ are not. Finally it is remarked that when the first class secondary constraints χΣ appear, the first class intrinsic constraints are not necessarily correct generators of the gauge transformations, but the correct ones can be expressed as linear combinations of the first class ΦA(or ΦB) and χΣ.