Home / Math / Recurgent Field Equation and Lagrangian Mechanics

Overview

This section develops the Recurgent Field Equation as the fundamental dynamical law governing semantic evolution, synthesizing the geometric foundations from §3, recursive stress-energy from §4, and semantic mass from §5 into a unified field theory. The equation emerges from a variational principle based on the Recurgent Lagrangian $\mathcal{L}$, which encodes the interplay between coherence propagation, recursive coupling, and autopoietic potential.

Key mathematical structures:

  1. Field Lagrangian $\mathcal{L} = \frac{1}{2}g^{ij}(\nabla_i C_k)(\nabla_j C^k) - V(C) + \Phi(C) - \lambda\mathcal{H}[R] + \mu W$
  2. Euler-Lagrange equations yielding the Recurgent Field Equation
  3. Noether currents preserving semantic symmetries

The theory posits that semantic evolution proceeds by minimizing the action integral $S = \int_{\mathcal{M}} \mathcal{L} \, dV$, subject to:

  • Conservation of recursive stress-energy (from §4)
  • Curvature-mass coupling (from §5)
  • Autopoietic potential thresholds
  • Wisdom field constraints

The dynamical law of RFT is now fully specified: semantic fields evolve by extremizing an action that weaves together coherence propagation, recursive tension, constraint geometry, and emergent wisdom. This variational framework yields field equations with intricate nonlinear behavior—admitting soliton-like solutions that anchor stable semantic structures, as well as phase transitions driven by autopoietic regeneration.

6.1 Lagrangian Density

The Lagrangian density $\mathcal{L}$ on the semantic manifold $\mathcal{M}$ is defined to encode the dynamics of the coherence field, recurgent generativity, and constraint enforcement:

\[\mathcal{L} = \frac{1}{2} g^{ij} (\nabla_i C_k)(\nabla_j C^k) - V(C_{\mathrm{mag}}) + \Phi(C_{\mathrm{mag}}) - \lambda \cdot \mathcal{H}[R]\]

where:

  • $C_k(p, t)$: Coherence vector field at point $p$ and time $t$,
  • $C_{\mathrm{mag}}(p, t) = \sqrt{g^{ij}(p, t) C_i(p, t) C_j(p, t)}$: Scalar coherence magnitude,
  • $V(C_{\mathrm{mag}})$: Attractor potential stabilizing the coherence field,
  • $\Phi(C_{\mathrm{mag}})$: Autopoietic recurgent potential,
  • $\mathcal{H}[R]$: Humility constraint functional, penalizing excessive recurgence,
  • $\lambda$: Humility weight parameter.

The first term represents the kinetic energy associated with spatial gradients of the coherence field. The remaining terms are scalar potentials, each a function of the coherence magnitude and recursive coupling, and serve to stabilize, generate, or constrain the evolution of the field.

Remark on Real and Complex Coherence Fields: The Lagrangian above is formulated for a real coherence field $C_i$. For systems exhibiting phase dynamics (e.g., solitonic solutions, see Section 7.10.1), a complexified Lagrangian is employed: \(\mathcal{L}_C = \frac{1}{2} g^{ij} (\nabla_i C_k)(\nabla_j C^{k*}) - V(C_{\mathrm{mag}}) + \Phi(C_{\mathrm{mag}}) - \lambda \cdot \mathcal{H}[R]\) where $C^{k}$ denotes the complex conjugate of $C^k$ and $C_{\mathrm{mag}} = \sqrt{g^{ij} C_i C_j^}$. This extension is required for describing wave-like and phase-dependent recurgent phenomena.

6.2 Action Principle

The action functional is given by

\[S = \int_{\mathcal{M}} \mathcal{L} \, dV\]

The system’s dynamics follow from the principle of stationary action: physical evolution corresponds to stationary points of $S$ under admissible variations, subject to the imposed constraints.

6.3 Euler–Lagrange Field Equation

Variation of the action with respect to $C_i$ yields the Euler–Lagrange equation:

\[\frac{\delta \mathcal{L}}{\delta C_i} - \nabla_j \left( \frac{\delta \mathcal{L}}{\delta (\nabla_j C_i)} \right) = 0\]

which, for the Lagrangian above, takes the explicit form

\[\Box C^i + \frac{\partial V(C_{\mathrm{mag}})}{\partial C_i} - \frac{\partial \Phi(C_{\mathrm{mag}})}{\partial C_i} + \lambda \cdot \frac{\partial \mathcal{H}[R]}{\partial C_i} = 0\]

where

  • $\Box = \nabla^a \nabla_a$ is the covariant d’Alembertian (semantic Laplacian).

The derivatives of the scalar potentials with respect to the vector field components are computed via the chain rule:

\[\frac{\partial V(C_{\mathrm{mag}})}{\partial C_i} = \frac{dV}{dC_{\mathrm{mag}}} \cdot \frac{\partial C_{\mathrm{mag}}}{\partial C_i} = \frac{dV}{dC_{\mathrm{mag}}} \cdot \frac{g^{ij} C_j}{C_{\mathrm{mag}}}\] \[\frac{\partial \Phi(C_{\mathrm{mag}})}{\partial C_i} = \frac{d\Phi}{dC_{\mathrm{mag}}} \cdot \frac{\partial C_{\mathrm{mag}}}{\partial C_i} = \frac{d\Phi}{dC_{\mathrm{mag}}} \cdot \frac{g^{ij} C_j}{C_{\mathrm{mag}}}\]

The humility constraint term involves a more intricate dependence, as $R$ is a functional of $C$ via the underlying semantic field $\psi$:

\[\frac{\partial \mathcal{H}[R]}{\partial C_i} = \int_{\mathcal{M}} \frac{\delta \mathcal{H}[R]}{\delta R_{jkl}(s, t)} \cdot \frac{\delta R_{jkl}(s, t)}{\delta C_i(p)} \, dV_s\]

The final term thus encodes the indirect coupling between $C_i$ and $R_{jkl}$, mediated by $\psi$.

Given the evolution equation for $R$, \(\frac{dR_{ijk}}{dt} = \Phi(C_{\mathrm{mag}}) \cdot \chi_{ijk},\) the humility constraint $\mathcal{H}[R]$ introduces a nontrivial feedback mechanism, whereby the present state of coherence modulates the future structure of recursive coupling.

6.4 Structural Interpretation

The above formalism constitutes a semantic field theory structurally analogous to established physical field theories (e.g., general relativity, Yang–Mills):

  • The curvature term ($\Box$) governs the propagation of recursive structure,
  • The potentials ($V(C)$, $\Phi(C)$) define the landscape of stable and generative attractors,
  • The constraint ($\mathcal{H}$) regulates recursion.

The resulting theory describes the evolution of coherence under the combined influence of gradient flow, potential-driven dynamics, and constraint enforcement.

6.5 Coupled Field Dynamics

Although the Lagrangian and resulting field equations are expressed in terms of the coherence field $C_i$, a complete description also calls for explicit consideration of the underlying semantic field $\psi_i$ and its evolution.

6.5.1 Semantic Field Evolution

The semantic field $\psi_i$ evolves according to

\[\frac{\partial \psi_i(p, t)}{\partial t} = v_i(p, t)\]

where the semantic velocity field $v_i(p, t)$ is given by

\[v_i(p, t) = \alpha \cdot \nabla_i C_{\mathrm{mag}}(p, t) + \beta \cdot F_i(p, t) + \gamma \cdot \mathcal{R}_i[\psi](p, t)\]

with

  • $\alpha \cdot \nabla_i C_{\mathrm{mag}}(p, t)$: Gradient-driven flow toward regions of higher coherence,
  • $\beta \cdot F_i(p, t)$: Recursive force arising from the surrounding semantic mass,
  • $\gamma \cdot \mathcal{R}_i\psi$: Direct recursive feedback.

This establishes a bidirectional coupling:

  1. $\psi_i$ determines $C_i$ via the coherence functional,
  2. $C_i$ influences the evolution of $\psi_i$ through its gradient.

6.5.2 Full Dynamical System

The coupled system is thus:

\[\frac{\partial \psi_i(p, t)}{\partial t} = v_i(p, t)\] \[\Box C_i + \frac{\partial V}{\partial C_i} - \frac{\partial \Phi}{\partial C_i} + \lambda \cdot \frac{\partial \mathcal{H}}{\partial C_i} = 0\] \[C_i(p, t) = \mathcal{F}_i[\psi](p, t)\]

This system may be integrated numerically by updating $\psi_i$ and deriving $C_i$ at each time step, or, in certain analytical regimes, reformulated to eliminate $\psi_i$ in favor of a closed evolution for $C_i$.

6.5.3 Consistency of the Action Principle

For the variational structure to hold, variations in $C_i$ must correspond to admissible variations in $\psi_i$. This is formalized via constrained variation:

\[\delta C_i(p, t) = \int_{\mathcal{M}} \frac{\delta C_i(p, t)}{\delta \psi_j(q, t)} \, \delta \psi_j(q, t) \, dq\]

The action principle continues to apply when such constraints are incorporated, so the coupled evolution of $C_i$ and $\psi_i$ remains compatible with the variational structure of the theory.

Next: Autopoietic Function and Phase Transitions