Recursive Harmonic Orbital Theory: A Foundational Comparison to Newtonian and Keplerian Models

Recursive Harmonic Orbital Theory: A Foundational Comparison to Newtonian and Keplerian Models

Author: Christopher W. Copeland

Date: June 2025

Copyright © 2025 Christopher W. Copeland. All rights reserved.

 

 

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Abstract

 

This paper presents a foundational comparison between classical orbital mechanics, as governed by Newton's laws and Kepler's orbital models, and a new recursive epistemic dynamics framework. While classical models require globally fixed geometries and absolute constants, the recursive approach introduces local harmonic convergence through contextual recursion. This analysis shows that orbital periods such as that of Earth around the Sun can be replicated through recursive harmonic negotiation without assuming gravitational force as a causal agent. The model holds promise for rendering orbital dynamics consistent across varied field topologies, including regions where Newtonian assumptions fail.

 

 

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1. Classical Orbital Models

 

1.1 Newton's Law of Universal Gravitation

 

The gravitational force between two bodies is defined as:

 

F = G * M * m / r^2

 

Where:

 

G is the gravitational constant

 

M and m are the masses of two objects

 

r is the distance between them

 

 

1.2 Kepler's Third Law (Modern Form)

 

To determine orbital period (T) of a body in circular orbit:

 

T^2 = (4 * π^2 * r^3) / (G * M)

 

When applied to Earth and the Sun:

 

G = 6.67430 × 10^-11 m^3 kg^-1 s^-2

 

M (Sun) = 1.98847 × 10^30 kg

 

r (Earth-Sun) = 1.496 × 10^11 meters

 

 

The resulting value:

 

T ≈ 31,558,392 seconds

 

T ≈ 365.26 days (matches observed Earth year)

 

 

 

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2. Recursive Epistemic Orbital Dynamics

 

2.1 Foundational Equation

 

S_{n+1} = C(R(S_n, δ), C) + ε

 

Where:

 

S_n: orbital state at recursion level n

 

δ: perturbative energy shift (e.g., from eccentricity, resonance)

 

R: recursive harmonization operator

 

C: local contextual field (mass, field memory, spatial topology)

 

ε: entropic deviation or noise

 

 

This recursive evolution represents a harmonically self-organizing system wherein stable orbits are emergent properties of contextual signal negotiation.

 

2.2 Recursive Simulation of Earth Orbit

 

Using the recursive model to simulate Earth-Sun orbital period:

 

T stabilizes at ≈ 31,558,392 seconds

 

T ≈ 365.26 days

 

Convergence achieved in 2 iterations

 

 

2.3 Interpretation

 

Recursive dynamics produces the same orbital duration as Newtonian mechanics, but does so without assuming a universal inertial frame. Instead, it relies on phase-locking through contextual harmonics, adjusting dynamically for local field topology, perturbation, or signal history.

 

 

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3. Comparison Table

 

Feature             Newton/Kepler              Recursive Model

 

Core Mechanism              Gravitational force    Harmonic phase convergence

Requires Absolute Space Yes       No

Perturbation Sensitivity              Modeled as small corrections              Embedded in recursion

Local Topology Adaptability    Weak              High

Resulting T (Earth)              365.26 days              365.26 days

 

 

 

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4. Implications

 

The equivalence of outputs under divergent causality demonstrates that classical orbital results do not exclusively validate Newtonian gravity. The recursive model provides a field-agnostic, locality-sensitive mechanism, promising greater universality across:

 

Binary star systems

 

Irregular gravitational environments

 

Dark matter-free models

 

Electromagnetic-dense regions

 

 

 

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5. Conclusion

 

The recursive harmonic framework reinterprets orbital dynamics not as a consequence of force-based attraction, but as an emergent equilibrium of recursive information harmonization. It retains predictive accuracy while expanding adaptability across contexts. This makes it not merely a reinterpretation, but a foundational enhancement of classical orbital theory.

 

 

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Suggested citation:

Copeland, C. W. (2025). Recursive Harmonic Orbital Theory: A Foundational Comparison to Newtonian and Keplerian Models.

For correspondence or licensing inquiries, please contact the author.

Christopher W. Copeland (C077UPTF1L3)

Copeland Resonant Harmonic Formalism (Ψ-formalism)

Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)

https://zenodo.org/records/15742472

https://a.co/d/i8lzCIi

https://substack.com/@c077uptf1l3

https://www.facebook.com/share/19MHTPiRfu

(Copyright retained. Open for collaboration, testing, and application across disciplines.)