Dynamical Structure of Irregular Constrained Systems

TL;DR

This paper analyzes irregular Hamiltonian systems with dependent constraints, classifying their behavior near the constraint surface, and discusses implications for regularization, equivalence of formalisms, and applications to higher-dimensional theories.

Contribution

It introduces a classification of irregular constrained systems into two types and explores their regularization and dynamical properties, especially regarding their equivalence and implications for quantization.

Findings

Type I constraints can be regularized for equivalent Hamiltonian and Lagrangian descriptions.

Type II constraints may be irregular and lead to inequivalence between formalisms.

Irregularities affect linearized theories and quantization, with relevance to higher-dimensional Chern-Simons theories.

Abstract

Hamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the constraint surface into two fundamental types. If the irregular constraints are multilinear (type I), then it is possible to regularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent. When the constraints are power of a linear function (type II), regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. It is shown that the inequivalence between the two formalisms can occur if the kinetic energy is an indefinite quadratic form in the velocities. It is also shown that a system of type I can evolve in time from a regular configuration into an irregular one, without any catastrophic changes. Irregularities have important consequences in the linearized approximation to nonlinear theories, as well as for the quantization of such systems. The relevance of these problems to Chern-Simons theories in higher dimensions is discussed.