Involution and constrained dynamics. I. The Dirac approach

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.

Stable Underlying Equations for Constrained Hamiltonian Systems

“…We must then check whether all secondary constraints are preserved by repeating the procedure until we either encounter case (i) or all constraints lead to case (ii). This is the famous Dirac algorithm, a special version of the general completion procedure for differential algebraic equations [30,32].…”

Deducing the Constraints in the Light-Cone SU(3) Yang-Mills Mechanics Via Gröbner Bases

“…It is worth to note here that the described method to find constraints within the Dirac formalism represent the reformulation of completion of the initial Hamiltonian equations to involution in another words and constraints corresponds to a set of the integrability conditions [18,19,20]. Now the algorithmic reformulation of the above stated scheme will be described using the ideas and the terminology of the Gröbner bases theory.…”

“…However, the further analysis of the consistency conditions (19) represents not so easy tractable issue. First of all, the number of Lagrange multipliers that can be determined from (19) depends on the rank of the structure group.…”

“…The consistency condition (19) for the "orthogonal" constraints χ a i ⊥ allows to determine the corresponding four Lagrange multiplier V ⊥ (τ ) and therefore summarizing, the SU (2) light-cone Yang-Mills mechanics possesses 8 functionally independent first-class constraints ϕ …”

“…The consistency conditions (19) allow to find the corresponding Lagrange multipliers V a k ⊥ and to get the expressions modulo primary constraints…”