Dynamical Systems with First- and Second-Class Constraints and the Second Noether Theorem

TL;DR

This paper analyzes dynamical systems with first- and second-class constraints, clarifies their structure, and develops a method to construct symmetry generators, emphasizing the role of first-class constraints in local symmetries.

Contribution

It introduces a systematic approach to classify and separate constraints in Hamiltonian formalism and constructs symmetry generators considering the second Noether theorem.

Findings

Second-class constraints do not affect local symmetry transformations.

A canonical set of constraints simplifies the analysis of dynamical systems.

The method clarifies the structure of second-class constraints in the Dirac formalism.

Abstract

Dynamical systems, described by Lagrangians with first- and second-class constraints, are investigated. In the Dirac approach to the generalized Hamiltonian formalism, the classification and separation of the first- and second-class constraints are presented with the help of passing to an equivalent canonical set of constraints. The general structure of second-class constraints is clarified. The method of constructing the generator of symmetry transformations in the second Noether theorem is given. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry which entirely is entirely stipulated by all the first-class constraints.