Appendix E — Derivation 5: Unified Action Principle

Overview

In classical mechanics and field theory, physical evolution is derived by extremising an action—typically a functional of energy, curvature, or field configuration. In modal dynamics, the action principle is retained, but its content is fundamentally different.

The action is not built from energy, momentum, or spacetime geometry. It is a measure of coherence preservation—a structural cost associated with maintaining a mode’s internal phase while interacting with a non-uniform modal environment.

This derivation formulates the unified action principle for all modal evolution.

Let a mode be described by a complex coherence function:

ψ (x, t) = ρ (x, t) e^{i ϕ (x, t)}

We define the instantaneous anchoring cost density as:

L (ψ, \partial_{t} ψ, \nabla ψ) = γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}

Where:

$γ$ is the temporal stiffness (penalising rapid phase change)
$α$ is the spatial stiffness (penalising high phase curvature)
$β$ is the anchoring penalty (penalising unsupported structure)

This replaces both the classical Lagrangian and field-theoretic energy functionals.

2. Full Action

The total action for a mode over time is:

S [ψ] = \int L (ψ, \partial_{t} ψ, \nabla ψ) d^{3} x d t

This is a four-dimensional cost—the integrated tension of preserving modal structure across time and space.

3. Variational Derivation

We extremise the action with respect to small variations $δ ψ^{*}$ , assuming fixed boundary conditions:

δ S [ψ] = 0

Using the Euler–Lagrange equation for complex fields:

\frac{δ S}{δ ψ^{*}} = - γ \frac{\partial^{2} ψ}{\partial t^{2}} + α \nabla^{2} ψ - β ψ = 0

This yields the modal evolution equation:

γ \frac{\partial^{2} ψ}{\partial t^{2}} = α \nabla^{2} ψ - β ψ

(See also Appendix A.)

4. Motion as Gradient Response

The position of a mode (e.g., an anchored cluster) evolves to minimise total cost. Let $\vec{x} (t)$ represent the mode’s effective centre of coherence. Then:

\frac{d^{2} \vec{x}}{d t^{2}} \propto - \nabla C

Where $C$ is the integrated anchoring cost in the surrounding field.

This replaces Newton’s second law. Motion becomes a bias-following gradient descent through coherence tension.

5. Interaction and Multi-Mode Action

For multiple modes $ψ_{i} (x, t)$ sharing the same space, the total action becomes:

S_{total} = \sum_{i} S [ψ_{i}] + \int Γ ({ψ_{i}}) d^{4} x

Where $Γ$ represents interference terms—structural cross-cost due to modal overlap, anchoring competition, or coherence saturation.

This term encodes:

Interaction (mode coupling through coherence conflict)
Exclusion (e.g., Pauli-like structural suppression)
Decay pathways (instability from cost divergence)

6. Summary of the Unified Action

The modal action:

S [ψ] = \int (γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}) d^{4} x

governs:

Time evolution (via $\partial_{t} ψ$ )
Spatial form (via $\nabla ψ$ )
Stability and decay (via $β | ψ |^{2}$ )
Motion and structure formation (via cost gradients)
Interaction and binding (via shared $Γ$ terms)

This is not an imposed rule—it is the single variational structure from which all modal behaviour emerges.

Conclusion

In modal dynamics, there are no separate force laws, field equations, or quantum rules.
There is one action. One principle. One cost functional.
Everything else follows.

Appendix D | [Index](./Appendix Master) | Appendix F