Matrix bordering structure of the Faddeev-Jackiw algorithm: Schur complement regularization and symbolic automation

TL;DR

This paper reveals that the Faddeev-Jackiw reduction process for singular Lagrangian systems can be understood through the Matrix Bordering Technique, with the Schur complement determining the termination condition and enabling symbolic automation.

Contribution

It establishes a geometric and algebraic link between the Faddeev-Jackiw algorithm and the Matrix Bordering Technique, providing a basis for symbolic automation of the reduction process.

Findings

The nondegeneracy of the extended matrix is governed by the Schur complement.

The Faddeev-Jackiw algorithm terminates for second-class constraint systems.

A symbolic Wolfram Language implementation automates the reduction process.

Abstract

We show that the iterative Faddeev-Jackiw (FJ) reduction for singular Lagrangian systems constitutes a geometrically constrained instance of the Matrix Bordering Technique (MBT). For a first-order Lagrangian with singular pre-symplectic form, each iteration of the Barcelos-Neto-Wotzasek algorithm produces an extended symplectic matrix of canonical bordered form, \begin{eqnarray} f^{(m)} = \left( \begin{matrix} f^{(0)} & B \\ -B^{\mathsf{T}} & 0 \end{matrix} \right) \end{eqnarray} where the bordering block $B$ is determined by the gradients of the consistency constraints. We prove that the nondegeneracy of the extended matrix is governed by the corresponding Schur complement, which is algebraically isomorphic to the Poisson bracket matrix of constraints. As a consequence, the Faddeev-Jackiw algorithm terminates if and only if the constraint algebra is nondegenerate, i.e., when the constraints form a second-class system. This algebraic characterization provides a rigorous foundation for automating the Faddeev-Jackiw procedure symbolically. We present a fully symbolic implementation in the Wolfram Language, and validate the approach on representative mechanical systems with nontrivial constraint structure. The resulting rule-based engine preserves parametric dependencies throughout the reduction, enabling reliable analysis of degeneracy, structural stability (when no bifurcations occur), and possible bifurcation scenarios as critical parameters are varied.