Understanding Temporal Flow Physics and the Relational Action Principle

Introduction

Temporal Flow Physics (TFP) provides a novel way to understand how systems evolve based on relational flows. Instead of focusing on absolute quantities, TFP models reality as chains of flows where each element interacts with its neighbors. This framework naturally leads to a variational principle and, ultimately, Euler-Lagrange equations.

TFP Local Flow Principle

In TFP, each flow f_i along a chain seeks to minimize local tension with its immediate neighbors:

\( V_i(f_i) = |f_i - f_{i-1}| + |f_i - f_{i+1}| \)

Equivalently, this defines a local discrete action functional:

\( S_i[f] = |f_i - f_{i-1}| + |f_i - f_{i+1}| \)

The local extremum condition is given by:

\( \frac{\partial S_i}{\partial f_i} = 0 \)

Chain-wide Dynamics

Summing over the entire chain gives the total action:

\( S[f] = \sum_i S_i[f] = \sum_i (|f_i - f_{i-1}| + |f_i - f_{i+1}|) \)

Varying \( f_i \) now produces the discrete Euler-Lagrange equations:

\( \frac{\partial}{\partial f_i} \sum_j |f_j - f_{j+1}| = 0 \)

This expresses how the system globally minimizes total local tension.

Continuum Limit

As the spacing between chain elements becomes infinitesimal (\( \Delta x \to 0 \)) and \( f_i \to f(x) \), differences become derivatives:

\( f_{i+1} - f_i \approx \Delta x \frac{df}{dx}, \quad |f_{i+1} - f_i| \approx \Delta x \left| \frac{df}{dx} \right| \)

The total action in the continuum is an integral:

\( S[f] = \int \mathcal{L}(f, f') \, dx, \quad \mathcal{L}(f, f') = |f'| \)

Euler-Lagrange Equations

The standard Euler-Lagrange equation for a functional \( S[f] = \int \mathcal{L} \, dx \) is:

\( \frac{d}{dx} \left( \frac{\partial \mathcal{L}}{\partial f'} \right) - \frac{\partial \mathcal{L}}{\partial f} = 0 \)

For \( \mathcal{L} = |f'| \):

\( \frac{\partial \mathcal{L}}{\partial f'} = \text{sign}(f'), \quad \frac{d}{dx}(\text{sign}(f')) = 0 \)

This expresses the continuum version of TFP flow equilibrium: the derivative of the flow gradient is constant, and flows settle to minimize local tension globally.

Including a Global Constraint

If we want to enforce a global invariant such as \( \Sigma = B_1 \times B_2 \), we introduce a Lagrange multiplier \( \lambda \):

\( S[f] = \int \mathcal{L}(f, f') \, dx + \lambda (\Sigma[f] - \Sigma_0) \)

Varying this action gives Euler-Lagrange equations subject to the invariant:

\( \frac{d}{dx} \left( \frac{\partial \mathcal{L}}{\partial f'} \right) - \frac{\partial \mathcal{L}}{\partial f} + \lambda \frac{\delta \Sigma}{\delta f} = 0 \)

This is the **Relational Action Principle**: a generalization of least action for systems of relational flows, derived directly from TFP.

Summary

• Local TFP dynamics minimize tension between neighboring flows (discrete variational principle).
• Summing local tensions produces a discrete Euler-Lagrange equation for the entire chain.
• Continuum limit produces a standard Euler-Lagrange equation with Lagrangian |f'|.
• Global invariants like Σ can be included using Lagrange multipliers, forming a Relational Action Principle.
• This shows how TFP naturally generates the mathematics of variational physics from purely relational principles.