Involution and Constrained Dynamics I: The Dirac Approach

TL;DR

This paper explores the application of involution theory to constrained systems, demonstrating that the Dirac algorithm completes equations to an involutive form and offering a more flexible analysis for field theories.

Contribution

It establishes a connection between the Dirac approach and involution theory, providing intrinsic formulas for degrees of freedom and extending the analysis to field theories.

Findings

Dirac algorithm completes equations to involutive systems

Involution analysis is more general than Dirac's method

Derived intrinsic expressions for degrees of freedom

Abstract

We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the Dirac approach. We also derive intrinsic expressions for the number of degrees of freedom.