Some properties of constraints in theories with degenerate Lagrangians

The Poisson brackets of primary constraints are expressed by means of linear differential operators in terms of Lagrangian constraints. A criterion for the existence in a theory of second-class constraints is proposed in the framework of the Lagrangian formalism. The Poisson brackets of primary constraints with the canonical Harniltonian are calculated. By means of Noether's theorem, Part II, it is shown that invariance of the action with respect to transformations with arbitrary functions of the time leads to primary constraints that are in involution with one another and with the canonical Hamiltonian, at least in the weak sense. It follows from the analysis of the functional arbitrariness in the solutions of the Hamilton equations that such primary constraints must be first-class constraints.