# Dirac-Bergmann Constraints in Physics: Singular Lagrangians, Hamiltonian Constraints and the Second Noether Theorem

## TL;DR

This paper reviews the mathematical properties of systems with singular Lagrangians, focusing on Dirac-Bergmann constraints, the connection to Noether's second theorem, and canonical transformations for gauge variables and observables.

## Contribution

It provides a comprehensive review of the mathematical structure of singular Lagrangian systems, linking constraints, Noether identities, and canonical transformations.

## Key findings

- Connection between Hessian eigenvalues and constraints
- Relation of Noether identities to Hamiltonian constraints
- Use of Shanmugadhasan transformation for gauge variables

## 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. Review paper for a chapter of a future book.