Dirac–Bergmann constraints in physics: Singular Lagrangians, Hamiltonian constraints and the second Noether theorem

Abstract

There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac–Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed:(i)the connection of the rank and eigenvalues of the Hessian matrix in the Euler–Lagrange equations with the chains of first- and second-class constraints;(ii)the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints;(iii)the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dirac observables, i.e. the quantities invariant under Hamiltonian gauge transformations.