Matrix bordering structure of the Faddeev-Jackiw algorithm: kernel reduction and symbolic automation

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). Since the initial pre-symplectic matrix is singular, the classical Schur determinantal identity cannot be applied. Instead, regularity is governed by the interaction of the bordering block with the kernel of the initial matrix. Writing N for an orthonormal basis of this kernel, we introduce the reduced constraint matrix $\Gamma =N^{T}B$ and derive an exact determinant factorization showing that the extended symplectic matrix is nondegenerate precisely when the generated constraints span the kernel and $\Gamma$ is nonsingular. We further identify $\Gamma$ with the Hessian of the symplectic potential restricted to the kernel and with the corresponding Faddeev-Jackiw constraint matrix. This establishes that, for independent generated constraints, termination of the FJ algorithm in a nondegenerate symplectic form is equivalent to nondegeneracy of the constraint algebra, characterizing a second-class system in the Dirac-Bergmann sense. These results provide the algebraic foundation for a fully symbolic implementation in the Wolfram Language. The resulting BorderedFJReduction engine preserves parametric dependencies throughout the reduction and is validated on representative constrained mechanical systems.

20 pages, 6 figures, Wolfram Language paclet available at https://github.com/echanlopez/BorderedFJReduction