TNFR Variational Principle — Lagrangian Action Formulation

Status: CANONICAL
Module: src/tnfr/physics/variational.py
Tests: tests/physics/test_variational.py (50 passing)
Date: 2026-03

1. Main Result

The TNFR nodal equation

$\frac{\partial \mathrm{EPI}}{\partial t} = \nu_f \cdot \Delta\mathrm{NFR}(t)$

admits a consistent variational formulation. It is the Euler-Lagrange equation of a well-defined action functional in the overdamped (dissipation-dominated) limit.

2. The TNFR Lagrangian

2.1 Lagrangian Density

On a graph $G=(V,E)$ the TNFR Lagrangian density at node $i$ is

$\mathcal{L}(i) \;=\; T(i) \;-\; V(i)$

where the transport (kinetic) energy and configuration (potential) energy are

$T(i) = \tfrac{1}{2}\bigl[J_\varphi(i)^{\,2} + J_{\Delta\mathrm{NFR}}(i)^{\,2}\bigr]$

$V(i) = \tfrac{1}{2}\bigl[\Phi_s(i)^{\,2} + |\nabla\varphi|(i)^{\,2} + K_\varphi(i)^{\,2}\bigr]$

2.2 Action Functional

$S_{\mathrm{TNFR}} = \int \mathrm{d}t \;\sum_{i\in V} \mathcal{L}(i,t)$

In discrete form (time steps of width $\Delta t$):

$S_{\mathrm{TNFR}} = \Delta t \;\sum_n \sum_i \mathcal{L}(i,\,t_n)$

2.3 Hamiltonian

The Legendre transform gives the TNFR Hamiltonian:

$\mathcal{H}(i) = T(i) + V(i) = \tfrac{1}{2}\bigl[\Phi_s^2 + |\nabla\varphi|^2 + K_\varphi^2 + J_\varphi^2 + J_{\Delta\mathrm{NFR}}^2\bigr]$

This is exactly $\tfrac{1}{2}\,\mathcal{E}(i)$ where $\mathcal{E}$ is the energy density from unified.compute_energy_density(), and the total Hamiltonian $H = \sum_i \mathcal{H}(i)$ equals conservation.compute_energy_functional().

3. Derivation of the Nodal Equation

3.1 Canonical Conjugate Pairs

The conservation law structure (Noether theorem, conservation.py) reveals two sectors with natural canonical pairs $(q,\,p)$:

3.2 Hamilton's Equations

$\frac{\partial q_i}{\partial t} = \frac{\partial \mathcal{H}}{\partial p_i} = p_i \qquad\text{(dynamics)}$

$\frac{\partial p_i}{\partial t} = -\frac{\partial \mathcal{H}}{\partial q_i} = -q_i \qquad\text{(force law)}$

3.3 Full Euler-Lagrange Equation

Using the EPI-ΔNFR pair with effective mass $m_i = 1/\nu_{f,i}$ (from classical_mechanics.py):

$\frac{1}{\nu_f}\,\frac{\partial^2\,\mathrm{EPI}}{\partial t^2} \;=\; -\frac{\partial V}{\partial \mathrm{EPI}}$

This has the same mathematical form as Newton's second law on the structural manifold: $F = m\,a$ with force $F = -\partial V/\partial\mathrm{EPI}$ and mass $m = 1/\nu_f$.

3.4 Overdamped Limit

The full TNFR dynamics includes structural dissipation (grammar-compliant evolution has $dH/dt \le 0$). Adding the dissipation term:

$\frac{1}{\nu_f}\,\frac{\partial^2\,\mathrm{EPI}}{\partial t^2} + \gamma\,\frac{\partial\,\mathrm{EPI}}{\partial t} = -\frac{\partial V}{\partial\mathrm{EPI}}$

In the overdamped limit ($1/\nu_f \to 0$ or $\gamma \to \infty$), the inertial term vanishes:

$\gamma\,\frac{\partial\,\mathrm{EPI}}{\partial t} = -\frac{\partial V}{\partial\mathrm{EPI}}$

Identifying $1/\gamma = \nu_f$ and $\Delta\mathrm{NFR} = -\partial V/\partial\mathrm{EPI}$:

$\boxed{\frac{\partial\,\mathrm{EPI}}{\partial t} = \nu_f \cdot \Delta\mathrm{NFR}(t)}$

Key derived result: Within this variational formulation, $\Delta\mathrm{NFR}$ is the negative functional gradient of the structural potential $V$.

4. Grammar Rules as Stationarity Conditions

Each grammar rule U1–U6 maps to a condition on the action functional $S_{\mathrm{TNFR}}$:

5. The 13 Operators as Canonical Transformations

In the Hamiltonian formulation, each operator acts as a transformation on the phase space $(q,\,p) = (\mathrm{EPI},\,\Delta\mathrm{NFR})$.

Grammar U2 in variational language: For every generating (expansive) operator, the sequence must include a dissipative operator to restore energy balance, ensuring $S < \infty$.

Dual-lever complement: The Hamiltonian classification above (Generating / Canonical / Dissipative) maps onto the experimentally observed dual-lever structure: operators act through the capacity lever ($\nu_f$) or the pressure lever ($\Delta$NFR), and the resulting $\Delta H$ sign follows from which lever is engaged. The Generating class corresponds predominantly to pressure-lever operators that inject $\Delta$NFR; the Dissipative class corresponds to pressure-lever operators that absorb it. See STRUCTURAL_OPERATORS.md §17.1 and example 39.

6. Thresholds as Critical Points of $V$

The TNFR potential $V(i) = \tfrac{1}{2}[x_1^2 + x_2^2 + x_3^2]$ is quadratic in each field. The canonical thresholds mark transition points where the effective potential (including grammar constraints as Lagrange multipliers) changes character:

At these values, the restoring force $-\partial V/\partial x = -x$ reaches the threshold intensity, and beyond them grammatically non-compliant dynamics sets in.

7. Interaction Lagrangian

The bilinear cross-sector coupling is:

$\mathcal{A}(i) = \Phi_s \cdot |\nabla\varphi| + K_\varphi \cdot J_\varphi + |\nabla\varphi| \cdot J_{\Delta\mathrm{NFR}}$

This equals unified.compute_action_density(). In the complete Lagrangian with interactions:

$\mathcal{L}_{\mathrm{full}} = T - V - \mathcal{A}$

The free Lagrangian $\mathcal{L} = T - V$ governs the independent sector dynamics, while $\mathcal{A}$ mediates the coupling between geometric and potential sectors.

8. Virial Theorem

At equilibrium (virialisation), the time-averaged kinetic and potential energies satisfy:

$\langle T \rangle = \langle V \rangle \quad\Longleftrightarrow\quad \text{virial ratio}\; T/V = 1$

• $T/V < 1$: potential-dominated (attractor basins, stable configurations)
• $T/V > 1$: kinetic-dominated (transport phase, active reorganisation)
• $T/V \approx 1$: near bifurcation (critical point of the action)

The virial ratio is a diagnostic for system state and proximity to bifurcation.

9. Consistency Relations

The variational module is fully consistent with the existing TNFR physics stack:

All relations verified numerically across Watts-Strogatz, Barabási-Albert, and Grid topologies (57 tests, 100% pass rate).

10. Single Source of Truth Architecture

The 6 canonical structural fields admit three physically meaningful decompositions — different projections of the same 6-dimensional field space. The implementation ensures a single computational path with no duplicated formulae:

unified.py                    (Layer 3 — RAW quadratic forms)
  ├── compute_energy_density   ℰ(i) = Φ_s² + |∇φ|² + K_φ² + J_φ² + J_ΔNFR²
  └── compute_action_density   𝒜(i) = bilinear coupling term
       │
       ├──► variational.py      H(i) = ½·ℰ(i),  T, V, ℒ = T−V
       │    (delegates via _raw_energy_density / _action_density)
       │
       └──► conservation.py     E = ½·Σℰ(i),  ρ, div(J), Q
            (delegates via _raw_energy_density)

10.1 Sector Decompositions

All three satisfy the consistency identity $T(i) + V(i) = \tfrac{1}{2}\,\mathcal{E}(i)$ at every node, verified by translate_sectors() with residual < $10^{-12}$.

10.2 translate_sectors() API

from tnfr.physics.variational import translate_sectors

result = translate_sectors(G)
# result['variational']        -> {'T': {...}, 'V': {...}}
# result['conservation']       -> {'rho': {...}, 'J_phi': {...}, 'J_dnfr': {...}}
# result['unified_psi']        -> {node: K_φ + i·J_φ}
# result['energy_density']     -> raw ℰ from unified.py
# result['consistency_check']  -> max |T+V − ½ℰ|  (should be ~0)

11. API Summary

from tnfr.physics.variational import (
    # Core densities
    compute_kinetic_density,       # T(i) = ½[J_φ² + J_ΔNFR²]
    compute_potential_density,     # V(i) = ½[Φ_s² + |∇φ|² + K_φ²]
    compute_lagrangian_density,    # ℒ(i) = T(i) − V(i)
    compute_hamiltonian_density,   # H(i) = ½·ℰ(i) — delegates to unified.py
    compute_interaction_density,   # 𝒜(i) — delegates to unified.compute_action_density
    
    # Sector translation
    translate_sectors,             # All three decompositions + consistency check
    
    # Phase space
    identify_conjugate_pairs,      # (K_φ,J_φ), (Φ_s,J_ΔNFR)
    compute_phase_space_volume,    # Liouville volume
    compute_poisson_bracket_estimate,
    
    # Snapshot & tracking
    capture_lagrangian_snapshot,    # Complete instant analysis
    compute_euler_lagrange_residual,
    compute_action_functional,     # S = ∫ℒ dt
    VariationalTracker,            # Time-series accumulation
    
    # Canonical checks
    check_symplectic_preservation,
    classify_operator_canonical,
    
    # Grammar stationarity
    analyze_grammar_stationarity,  # U1-U6 as δS conditions
    analyze_potential_critical_points,
    
    # Comprehensive
    compute_variational_suite,     # Full analysis in one call
)

Implementation & Examples

Executable Demonstrations

Key Source Modules

• src/tnfr/physics/variational.py — Lagrangian, Hamiltonian, Euler-Lagrange, symplectic checks
• src/tnfr/physics/conservation.py — Energy functional (Lyapunov candidate)
• src/tnfr/physics/classical_mechanics.py — Classical limit correspondence

References

• Nodal equation: TNFR.pdf §2.1, AGENTS.md §Foundational Physics
• Conservation theorem: STRUCTURAL_CONSERVATION_THEOREM.md
• Grammar rules: UNIFIED_GRAMMAR_RULES.md
• Classical mechanics analogy: src/tnfr/physics/classical_mechanics.py
• Unified fields (canonical source): src/tnfr/physics/unified.py
• Energy functional: src/tnfr/physics/conservation.py