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,
(m)
f

(0)
f
B
-B
0
,
​where the bordering block $B$ is determined by the gradients of the consistency constraints. Because
(0)
f
is singular by hypothesis, the classical Schur determinantal identity—which requires an invertible anchor block—cannot be invoked. The regularity of
(m)
f
is instead governed entirely by the interaction of
B
with
ker
(0)
f

. Writing
N
for an orthonormal basis of
ker
(0)
f

, we introduce the reduced constraint matrix
Γ
B
N
and prove the exact factorization
det
(m)
f

∏
ℓ
2
μ
ℓ
det
2
(Γ)
, where the
μ
ℓ
>0
are the Pfaffian factors of the nonsingular block of
(0)
f
. This yields the sharp nondegeneracy criterion:
(m)
f
is regular if and only if the number of generated constraints equals
dimker
(0)
f

and
Γ
is nonsingular. Our central algebraic result is that this reduced matrix coincides—coefficient by coefficient—with the Hessian of the symplectic potential restricted to
ker
(0)
f

,
Γ
αβ

i
N
a
∂
i
∂
j
V
j
N
b


Ω
α
,
Ω
β

FJ
,
​which is precisely the constraint matrix whose nondegeneracy characterizes a second-class system in the Dirac--Bergmann sense. As a consequence, we prove the biconditional
det
(m)
f
≠0 ⟺ det(Γ)≠0
: assuming the generated constraints are independent, the Faddeev-Jackiw algorithm terminates in a nondegenerate symplectic form 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. CITATION (original article): Chan–López, E., Martín–Ruiz, A., Cabrera, J. M. et al. (2026). Matrix bordering structure of the Faddeev–Jackiw algorithm: kernel reduction and symbolic automation. Eur. Phys. J. Plus, 141, 932. https://doi.org/10.1140/epjp/s13360-026-08146-x

Table of Contents

◼
  • GETTING STARTED
    • ◼
  • Installation and interface
    • ◼
  • Introduction
    • 1
    .
    Install from the latest GitHub release
    2
    .
    The interface
    ◼
  • THE REDUCTION AT WORK
    ​Mechanical systems and auxiliary functions
    • 3
    .
    A first reduction
    4
    .
    Applications
    4
    .
    1
    .
    Compound spring
    4
    .
    2
    .
    Pendulum and two springs
    4
    .
    3
    .
    Masses, springs and ring
    5
    .
    Auxiliary functions
    5
    .
    1
    .
    Faddeev–Jackiw matrix
    5
    .
    2
    .
    Structured visualization
    5
    .
    3
    .
    Subscripted formatting
    ◼
  • STRUCTURE AND LIMITS
    ​Candidates, diagnostics, migration
    • 6
    .
    Gauge candidates
    6
    .
    1
    .
    Branch 1: null contractions
    6
    .
    2
    .
    Branch 2: dependent constraints
    6
    .
    3
    .
    Why the verdict is a candidate
    7
    .
    Diagnostics, tracing and failure modes
    7
    .
    1
    .
    Per-pass trace
    7
    .
    2
    .
    Inconsistent systems
    7
    .
    3
    .
    Validated input
    8
    .
    Migration from v0.1.2
    ◼
  • References

    • Introduction

    Many mechanical systems are naturally described by Lagrangians whose kinetic structure is singular. In such systems, the canonical variables are not all independent, and the equations of motion are accompanied by consistency conditions that define constraints on phase space. The Faddeev--Jackiw (FJ) formalism provides a geometric way of handling these systems: instead of introducing constraints through an enlarged Hamiltonian formalism, it follows the degeneracy of the presymplectic form and incorporates the resulting consistency conditions iteratively.
    ​
    From a computational point of view, this procedure leads to a sequence of enlarged antisymmetric matrices. The structure of these matrices is particularly suited to symbolic computation because each reduction step can be interpreted as a matrix-bordering operation. BorderedFJReduction implements this structure directly in the Wolfram Language, allowing the Faddeev--Jackiw reduction to be carried out symbolically while retaining exact parameter dependence throughout the calculation.
    ​
    The mathematical viewpoint is equally important. The kernel of the presymplectic matrix identifies the directions along which the Lagrangian description is degenerate. Contracting these null directions with the gradient of the symplectic potential determines whether the degeneracy generates new consistency constraints or persists after the constraints already obtained have been imposed. In the latter situation, the reduction may signal a candidate for gauge symmetry. The present
    implementation deliberately distinguishes this structural identification from the subsequent construction and verification of gauge generators.
    ​
    This notebook illustrates these ideas through explicit mechanical systems. The examples are chosen to display different aspects of the reduction: noncanonical kinetic terms, coupled mechanical constraints, multiple constraints generated in a single bordering step, nonlinear and trigonometric constraints, and residual null directions associated with gauge-symmetry candidates. Particular attention is given to the symbolic structure returned by the paclet, so that the intermediate mathematical objects remain inspectable rather than being reduced to numerical output.
    ​
    The notebook is intended both as a computational companion to the published work and as an entry point for experimenting with the Faddeev--Jackiw formalism in the Wolfram Language.

    1. Install
    BorderedFJReduction
    from the Latest GitHub Release.

    The paclet can be installed directly from the GitHub release corresponding to the latest published version. The following command downloads and installs the
    BorderedFJReduction
    v0.1.3 paclet.
    ForceVersionInstall->True
    ensures that the specified release is installed even if another version is already present in the local paclet environment.
    In[]:=
    PacletInstall​​ "https://github.com/echanlopez/BorderedFJReduction/releases/download/v0.1.3/BorderedFJReduction-0.1.3.paclet",​​ ForceVersionInstallTrue​​
    Out[]=
    PacletObject
    Name: BorderedFJReduction
    Version: 0.1.3
    
    Confirm that the installed release is v0.1.3. Earlier releases report the retired verdicts
    "Found"
    /
    "NotFound"
    and count iterations differently, so the outputs below will not match.
    In[]:=
    PacletFind["BorderedFJReduction"]
    Out[]=
    PacletObject
    Name: BorderedFJReduction
    Version: 0.1.3
    
    Load the paclet. The splash panel reports the version, the build and the scope of the engine.
    In[]:=
    Needs["BorderedFJReduction`"]
    BorderedFJReduction
    A symbolic engine for the Faddeev–Jackiw reduction of singular Lagrangians,formulated as a geometrically constrained instance of matrix bordering.
    Scope: regularization, weak-vanishing constraint analysis, and structural identification of gauge candidates.
    Gauge generators are not constructed or verified by this engine.
    Authored by:
    Ramón Eduardo Chan López · José Alberto Martín Ruiz
    Jaime Manuel Cabrera · Jorge Mauricio Paulin Fuentes
    BorderedFJReduction v0.1.3
    Latest Release | Mathematica 15. | 28 August 2026

    2. The interface

    BorderedFJMatrix takes a first-order Lagrangian
    La.
    ′
    ξ
    -V
    and returns an opaque symbolic object. The information panel lists the signature, the options and the verdict vocabulary.
    In[]:=
    ?BorderedFJMatrix
    Out[]=
    Symbol
    BorderedFJMatrix[kinetic, potential, vars, opts] performs the symbolic Faddeev-Jackiw reduction of the first-order Lagrangian L = kinetic - potential over the phase-space variables vars, using geometrically constrained matrix bordering, and returns an opaque symbolic object queryable by key (see "Properties"). The recursion stops when the extended matrix is regularized, when an inconsistency is produced, or when it stalls on a kernel whose consistency contractions vanish weakly, in which case a GAUGE CANDIDATE is reported ("GaugeSymmetry" -> "CandidateFound"); no gauge generator is constructed or verified. BorderedFJMatrix[<|"kinetic-energy" -> ..., "symplectic-potential" -> ..., "vars" -> ...|>, opts] is equivalent. Options: "MaxIterations" (5, bounds bordering rounds) and "TraceStages" (False).
    A system is specified declaratively, by the three keys
    "kinetic-energy"
    ,
    "symplectic-potential"
    and
    "vars"
    . The kinetic term must be linear in the atomic velocities
    Derivative[1][q]
    ; the variables are atomic symbols, never functions of time, which is what allows the reduction to preserve parametric dependence exactly.
    In[]:=
    Options[BorderedFJMatrix]
    Out[]=
    {MaxIterations5,TraceStagesFalse}
    "MaxIterations"
    bounds the number of bordering rounds, that is the superscript m of the extended matrix
    (m)
    f
    ; it does not bound the number of passes, and the terminal readout pass is always granted.
    "TraceStages"
    records a per-pass trace under the key
    "StageData"
    (Section 6).

    3. A first reduction

    A singular Lagrangian with noncanonical kinetics. The one-form is
    a=(z,x^2/2,0)
    , so the pre-symplectic matrix is
    3×3
    antisymmetric and therefore singular: its kernel is one-dimensional and generates a single constraint.
    In[]:=
    ex1=​​<|"kinetic-energy"->z*Derivative[1][x]+x^2/2*Derivative[1][y],​​"symplectic-potential"->z^2/2+x^2/2y,​​"vars"->{x,y,z}|>;
    Since v0.1.3 the association may be passed directly. The idiom used in earlier versions of this notebook,
    BorderedFJMatrix@@Lookup[#,Keys@#]&
    , still works and is equivalent.
    In[]:=
    bfj1=BorderedFJMatrix[ex1]
    Out[]=
    BorderedFJMatrix
    ExtendedMatrixRank: 4
    GaugeSymmetry: NotCandidateFound
    MatrixStatus: Regularized
    ConsistencyCheck: True
    ConstraintsLength: 1
    OneFormLength: 4
    VariablesLength: 4
    IterationCount: 1
    PassCount: 2
    
    The summary box reports the rank of the extended matrix, the verdict, the terminal status, and the two counters. Open it to inspect the rest.
    In[]:=
    bfj1/@{"MatrixStatus","GaugeSymmetry","Constraints"}
    Out[]=
    Regularized,NotCandidateFound,-
    1
    2
    x(x+2z)
    The reduction regularized the form, so the generalized brackets exist and are the inverse of the extended matrix.
    In[]:=
    bfj1["InverseExtendedMatrix"]//MatrixForm
    Out[]//MatrixForm=
    0
    -
    1
    x
    0
    0
    1
    x
    0
    -
    x+z
    2
    x
    -
    1
    2
    x
    0
    x+z
    2
    x
    0
    -
    1
    x
    0
    1
    2
    x
    1
    x
    0
    Two invariants hold for every object returned by the engine and are worth using as an audit of the recursion: the number of bordering rounds equals the length of the per-stage record, and the total of that record equals the number of Lagrange multipliers introduced.
    In[]:=
    Length[bfj1["ConstraintsPerStage"]] === bfj1["IterationCount"]
    Total[bfj1["ConstraintsPerStage"]] === bfj1["MultiplierCount"]

    4. Applications

    4.1 Compound spring

    4.2 Pendulum and two springs

    4.3 Masses, springs and ring

    5. Auxiliary functions

    5.1 Faddeev–Jackiw matrix

    5.2 Structured visualization

    A composite system of four masses whose fourth momentum is a linear combination of the other three.

    5.3 Subscripted formatting

    ToSubscript is Listable, so it threads over matrices.

    6. Gauge candidates

    6.1 Branch 1: null contractions

    A ring-coupled variant of the previous composite system. Here every contraction of the residual kernel with the gradient of the potential vanishes identically, so no further constraint can be generated.
    The halting matrix is stored on every halt, so the candidate directions remain recoverable by the user even though the engine reports no generators.

    6.2 Branch 2: dependent constraints

    Here the candidates are not identically zero, but each of them lies in the ideal generated by the constraints already accumulated, so it vanishes on the constraint surface and restricts the phase space no further. Redundancy is decided by a Gröbner normal form, which is the weak-vanishing notion of Dirac consistency and is exact for nonlinear constraints; the gradient-rank test used up to v0.1.2 fails on data of this kind.
    The candidates discarded as weakly zero are kept, so the discarded material is auditable rather than lost.

    6.3 Why the verdict is a candidate and not a detection

    The following four-variable system is taken from E. Chan-López, "From Null Modes to Gauge Generators: A Bordered Faddeev–Jackiw Theory of Constraint Chains" (preprint), where it is established as a structural result rather than an illustration.

    7. Diagnostics, tracing and failure modes

    7.1 Per-pass trace

    The trace is diagnostic only: it never changes the verdict, and it is absent (an empty list) unless the option is set.

    7.2 Inconsistent systems

    7.3 Validated input

    The option itself is validated: it must be a positive integer or Infinity.

    8. Migration from v0.1.2

    Behaviour that changed silently.​
    Identically zero candidates are no longer adjoined as constraints; multipliers are numbered globally across rounds; redundancy is decided by ideal membership rather than by linear independence of gradients. Reductions whose v0.1.2 output contained spurious multipliers or duplicated constraints will therefore return smaller, correct systems.

    CITE THIS NOTEBOOK

    BorderedFJReduction: A Symbolic Engine for the Faddeev–Jackiw Reduction​
    by E. Chan-López, A. Martín-Ruiz, J. Manuel Cabrera and J. M. Paulin Fuentes
    Wolfram Community, STAFF PICKS, September 2, 2026
    ​https://community.wolfram.com/groups/-/m/t/3791192