The Limit of Noether Conserved Charges is the Number of Primary First-Class Constraints in a Constrained System

Abstract

We discuss the stationarity of generator G for gauge symmetries in two directions. One is to the motion equations defined by total Hamiltonian HT, and gives that the number of the independent coefficients in the generator G is not greater than the number of the primary first-class constraints, and the number of Noether conserved charges is not greater than that of the primary first-class constraints, too. The other is to the variances of canonical variables deduced from the generator G, and gives the variances of Lagrangian multipliers contained in extended Hamiltonian HE. And a second-class constraint generated by a first-class constraint may imply a new first-class constraint which can be combined by introducing other second-class constraints. Finally, we supply two examples. One with three first-class constraints (two is primary and one is secondary) has two Noether conserved charges, and the secondary first-class constraint is combined by three second-class constraints which are a secondary and two primary second-class constraints. The other with two first-class constraints (one is primary and one is secondary) has one Noehter conserved charge.