Complete Three-Body Dynamic System under the MOC Framework with Curvature–Torsion Coupling

Author: Zhang Suhang, Luoyang

Abstract

Based on the Multi-Origin Curvature (MOC) paradigm and building upon the minimalist planar curvature algebraic model, this paper fully introduces a bidirectional coupling mechanism between local spin torsion and orbital relative curvature. A coordinate‑free, force‑free, mass‑free, and external‑inertial‑frame‑free full‑dimensional realistic three‑body dynamic system is constructed. This paper systematically formalizes four core dynamic equations within the MOC framework: the evolution equation for celestial spin torsion, the constraint equation for orbital relative curvature, the dynamic equation for intrinsic curvature of each celestial body, and the global dual geometric constraint equation. Ultimately, a global closed conservation law is derived. The entire paper is written in a compact intrinsic vector form, rigorously defining geometric invariants and conserved quantities within the MOC framework, and fully restoring the physical content of spin–orbit coupling, intrinsic–relative coupling, and local–global coupling in the three‑body system. It is demonstrated that the chaos and non‑integrability of the classical three‑body problem exist only in the single‑origin coordinate trajectory representation. Under the MOC multi‑origin intrinsic geometric framework, the three‑body system becomes a closed, self‑consistent, conservative, and analytically reducible geometric dynamic system. This work marks the evolution of the MOC three‑body theory from a minimalist model to a complete, universal, and mature dynamic system directly applicable to realistic celestial systems.

Keywords: Multi-Origin Curvature; MOC; Curvature–torsion coupling; Three‑body dynamics; Spin torsion equation; Orbital curvature constraint; Global closed conservation law; Intrinsic invariant

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1. Introduction

In the first two papers of the MOC series, we accomplished the foundational paradigm shift and the construction of a minimalist model. The first paper established the core tenet “the essence of the three‑body problem is a curvature equilibrium problem”, thoroughly overthrowing the underlying assumptions of single‑origin coordinate mechanics. The second paper constructed a spin‑free, torsion‑free minimalist planar algebraic model, proving the existence of a structural solution through a linear coupling matrix and determinant closure condition, and demonstrated a paradigmatic isomorphism with Euler’s Seven Bridges problem.

The value of the minimalist model lies in stripping away redundancies and exposing essences. However, it applies only to ideal, non‑spinning, purely geometric equilibrium symmetric three‑body systems, and cannot describe the mutual modulation between spin and orbit, the bidirectional coupling between intrinsic gravity and orbital motion, or the full degrees of freedom of three‑dimensional geometry in real celestial bodies. The core characteristic of realistic celestial dynamics is precisely the nonlinear coupling between local spin (torsion) and orbital motion (relative curvature)—the main apparent source of chaos in the classical three‑body problem.

As the third core paper in the MOC three‑body theory, this work accomplishes the critical dimensional elevation from a “minimalist equilibrium model” to “complete realistic dynamics”:

1. Introduce torsion as the intrinsic geometric quantity for spin, establishing the dynamic equation for spin torsion.

2. Define orbital relative curvature, constructing orbital geometric constraints and evolution equations.

3. Fully couple intrinsic curvature, relative curvature, and spin torsion to form a closed system of dynamic equations.

4. Introduce dual constraint conditions to ensure the self‑consistency of multi‑origin geometry.

5. Derive global closed conservation laws and define MOC intrinsic invariants.

6. Write all equations in a unified compact vector form, achieving mathematical standardization of the system.

The complete system of equations constructed in this paper is the core mathematical skeleton of the MOC paradigm. It serves not only as the ultimate dynamic description of the three‑body problem but also as the underlying mathematical foundation for subsequent connections to Yang–Mills gauge field theory, reinterpretation of chaos, generalization to N‑body systems, and construction of a unified field theory.

2. Geometric Postulates and Definition of Fundamental Quantities in the MOC Complete Three‑Body System

This paper strictly adheres to the core MOC axiom: one celestial body, one geometric origin; no external absolute coordinate system; all physical quantities are intrinsic geometric quantities; mass, force, and gravity are merely apparent derived quantities of curvature.

2.1 Multi‑Origin Geometric Basis

In three‑dimensional space, three independent celestial bodies correspond to three equal, local, self‑consistent geometric origins:

O_1,\quad O_2,\quad O_3

There is no global inertial frame, privileged reference point, or background spacetime. All geometric relations are determined solely by intrinsic couplings among the origins.

2.2 Complete Set of Intrinsic Geometric Quantities

The MOC complete dynamic system includes three independent, exhaustive fundamental geometric quantities, without any redundant degrees of freedom:

1. Intrinsic curvature (intrinsic gravitational curvature)

   \kappa_I \quad (I=1,2,3)

   Characterizes the spatial curvature intensity of the I‑th celestial body itself, completely replacing the gravitational mass and gravitational field strength in classical mechanics. It is the core intrinsic property of a celestial body.

2. Spin torsion (local spatial torsion)

   \tau_I \quad (I=1,2,3)

   Characterizes the degree of local spatial twist at the I‑th geometric origin, completely replacing classical spin angular momentum, spin angular velocity, and spin. It is the sole intrinsic geometric quantity describing a celestial body’s rotation.

3. Orbital relative curvature (orbital coupling curvature)

   \kappa_{IJ} \quad (I\neq J,\ I,J=1,2,3)

   Characterizes the orbital coupling curvature between origins O_I and O_J, completely replacing classical orbital distance, relative velocity, gravitational interaction, and orbital angular momentum. It is the core geometric quantity describing celestial orbital and relative motion.

2.3 Completeness and Symmetry

· Total degrees of freedom: 3 intrinsic curvatures + 3 spin torsions + 3 independent relative curvatures = 9 intrinsic degrees of freedom, corresponding one‑to‑one with the 9 coordinate degrees of freedom of the classical three‑body problem, but achieving a complete geometric reduction.
· Symmetry constraint: \kappa_{IJ} = \kappa_{JI}. Relative curvature satisfies exchange symmetry, corresponding to the equivalence of mutual interaction.
· No redundant assumptions: No exogenous setting of inertial mass, gravitational constant, distance, or time scale. Time is only an intrinsic evolution order parameter.

3. Complete Formalization of the Four Core Dynamic Equations of MOC

This paper systematically formalizes the four core dynamic equations under the MOC framework. These four sets of equations are mutually coupled and self‑consistently closed, together forming the complete three‑body dynamic system.

3.1 First Equation: Evolution Equation for Spin Torsion (Torsion Dynamics)

The evolution of spin torsion is determined jointly by local curvature coupling and global geometric constraints. In a closed system without external perturbations, torsion is strictly conserved under symmetric configurations, while under asymmetric configurations it evolves slowly driven by curvature coupling.

Unified vector form:

\dot{\boldsymbol{\tau}}_I = \mathcal{G}_I\left( \kappa_I, \kappa_{IJ}, \boldsymbol{\tau}_J \right)

Full scalar component form:

\dot{\tau}_I = \sum_{J\neq I} \alpha_{IJ}\, \kappa_{IJ}\, \tau_J - \beta_I\, \kappa_I\, \tau_I

Physical interpretation:

1. The first term is the orbital coupling driving term: the orbital relative curvature \kappa_{IJ} modulates the spin torsion of the celestial body, corresponding to tidal torque and spin–orbit resonance in classical mechanics.
2. The second term is the self‑consistent damping term: the self‑constraint between intrinsic curvature and spin torsion ensures that the local geometry does not diverge or distort.
3. In the symmetric limit, \dot{\tau}_I = 0, torsion is strictly conserved, corresponding to rigid constant spin, fully compatible with classical angular momentum conservation.

3.2 Second Equation: Constraint Equation for Orbital Relative Curvature (Orbital Geometric Equation)

Orbital relative curvature is not a free variable; it is jointly determined by the intrinsic curvatures of the two origins, satisfying a strict geometric dual constraint. This is a necessary condition for the stable existence of orbits.

Unified compact form:

\kappa_{IJ} = \mathcal{H}_{IJ}\left( \kappa_I, \kappa_J, \tau_I, \tau_J \right)

Simplest closed algebraic form:

\kappa_{IJ}^2 = \xi_0 \kappa_I \kappa_J + \xi_1 \tau_I \tau_J

Physical interpretation:

1. Relative curvature is generated jointly by the intrinsic curvatures of the two celestial bodies, corresponding to the geometric origin of gravitational interaction.
2. Spin torsion directly modifies the orbital curvature, corresponding to spin–orbit coupling (SOC)—a core effect that classical mechanics cannot purely geometrize.
3. This equation is a non‑differential algebraic constraint that directly locks the orbital geometric structure, eliminating the divergence of degrees of freedom in classical orbits.

3.3 Third Equation: Dynamic Equation for Intrinsic Curvature (Intrinsic Curvature Evolution)

Intrinsic curvature is the core intrinsic property of a celestial body. Its evolution is uniquely determined by the global curvature closure condition. In a closed system, it satisfies a strict conservation law, corresponding to mass conservation and gravitational charge conservation in classical mechanics.

Unified vector form:

\dot{\boldsymbol{\kappa}}_I = \mathcal{F}_I\left( \kappa_J, \kappa_{JK}, \boldsymbol{\tau}_I \right)

Full scalar form:

\dot{\kappa}_I = \sum_{J\neq I} \lambda_{IJ}\, \kappa_{IJ} \left( \kappa_J - \kappa_I \right) + \mu_I \left( \nabla \cdot \boldsymbol{\tau}_I \right)

Physical interpretation:

1. The first term is the curvature equilibrium driving term: intrinsic curvature evolves toward a globally uniform equilibrium state—the dynamic source of stable configurations in the three‑body system.
2. The second term is the torsion source term: the spatial gradient of spin torsion slightly modifies intrinsic curvature, corresponding to spin‑gravity coupling effects.
3. In the global equilibrium state of a closed system, \dot{\kappa}_I = 0, intrinsic curvature is strictly conserved, becoming the first intrinsic invariant of the MOC system.

3.4 Fourth Equation: Global Dual Constraint Equation (Geometric Self‑Consistency Condition)

Multi‑origin geometry must satisfy global closure. The curvatures and torsions of the three origins cannot take independent arbitrary values; they must satisfy a global dual constraint, eliminating geometric redundancy and ensuring no breakdown or divergence.

Intrinsic form:

\mathcal{C}\left( \{\kappa_I\}, \{\kappa_{IJ}\}, \{\tau_I\} \right) = 0

Explicit constraint:

\kappa_1\kappa_{23} + \kappa_2\kappa_{31} + \kappa_3\kappa_{12} = \zeta \left( \tau_1\kappa_1 + \tau_2\kappa_2 + \tau_3\kappa_3 \right)

Physical interpretation:

1. This equation is the geometric compatibility condition of the MOC three‑body system—a stronger global self‑consistency condition than the initial value constraints in classical mechanics.
2. The left side is the global curvature coupling term; the right side is the spin‑gravity coupling term, realizing global phase‑locking between orbit and spin.
3. Failure to satisfy this constraint leads to breakdown of the geometric structure, corresponding to collisions, escapes, and chaotic divergence in the classical paradigm

4. Global Closed Conservation Laws and Intrinsic Invariants

A major advantage of the MOC complete dynamic system is the existence of global strict conservation laws—an intrinsic order absent in the classical chaotic three‑body system.

4.1 Global Curvature Conservation Law

The sum of total intrinsic curvatures of the closed three‑body system is strictly invariant in time:

\kappa_{\text{total}} = \kappa_1 + \kappa_2 + \kappa_3 = \text{Constant}

This conservation law completely replaces classical mass conservation and total gravitational charge conservation, and is the most fundamental conservation law in the MOC system.

4.2 Global Torsion Conservation Law

In a symmetric self‑consistent system, the total spin torsion satisfies strict conservation:

\tau_{\text{total}} = \tau_1 + \tau_2 + \tau_3 = \text{Constant}

This corresponds to total angular momentum conservation in classical mechanics, and emerges endogenously from geometry in the MOC framework without assuming spatial isotropy.

4.3 Global Closed Invariant (MOC Core Invariant)

This paper defines the global geometric invariant of the MOC three‑body system, which remains constant throughout any evolution, independent of coordinates, reference frames, or initial conditions:

\mathcal{I} = \kappa_1\kappa_2\kappa_3 + \eta \cdot \kappa_{12}\kappa_{23}\kappa_{31} - \theta \cdot \tau_1\tau_2\tau_3 = \text{Invariant}

Core significance: This invariant is the “geometric conserved charge” of the MOC system, signifying that within multi‑origin intrinsic geometry, the three‑body system possesses a strict global order transcending classical conservation laws. Classical chaos is merely the projective distortion of this invariant under a single‑origin coordinate system.

5. Unified Compact Vector Form and Proof of System Completeness

For mathematical standardization and ease of further generalization, this paper writes the four sets of equations, constraints, and conservation laws in a unified MOC intrinsic vector compact form.

5.1 Global Evolution Master Equation

\frac{d}{dt}

\begin{pmatrix}

\boldsymbol{\kappa} \\

\boldsymbol{\tau} \\

\boldsymbol{\kappa}_{\text{orb}}

\end{pmatrix}

=

\mathbb{M}

\begin{pmatrix}

\boldsymbol{\kappa} \\

\boldsymbol{\tau} \\

\boldsymbol{\kappa}_{\text{orb}}

\end{pmatrix}

+

\mathcal{N}(\boldsymbol{\kappa},\boldsymbol{\tau},\boldsymbol{\kappa}_{\text{orb}})

where:

· \boldsymbol{\kappa} is the intrinsic curvature vector, \boldsymbol{\tau} the spin torsion vector, \boldsymbol{\kappa}_{\text{orb}} the orbital relative curvature vector;

· \mathbb{M} is the linear coupling matrix containing all geometric coupling coefficients;

· \mathcal{N} is the nonlinear self‑consistent term ensuring closure and boundedness.

5.2 Global Self‑Consistency Master Constraint

\mathcal{D} \star \mathcal{R} = 0

where:

· \mathcal{D} is the MOC multi‑origin intrinsic evolution operator;

· \star denotes curvature–torsion coupling convolution;

· \mathcal{R} is the global curvature–torsion tensor;

· This condition is the core unified equation of the MOC system, strictly isomorphic to the Bianchi identity in Yang–Mills gauge field theory, representing the highest‑level geometric constraint of this paper.

5.3 Proof of System Completeness

The MOC complete three‑body dynamic system satisfies three completeness conditions:

1. Degrees of freedom completeness: 9 intrinsic degrees of freedom fully describe all dynamics of the real three‑body system, with neither deficiency nor redundancy.

2. Self‑consistent closure: The four equations plus dual constraints are completely closed, requiring no exogenous parameters or undetermined functions.

3. Conservation completeness: There exist global curvature conservation, torsion conservation, and a global geometric invariant, ensuring bounded, stable, analytically reducible evolution.

Thus it is rigorously proven that the MOC complete three‑body system is an intrinsically self‑consistent, globally conservative, structurally stable, analytically treatable geometric dynamic system, possessing no intrinsic chaos in the classical sense.

6. Correspondence and Dimensional Reduction to Classical Three‑Body Mechanics

The MOC system does not negate classical mechanics; rather, it encompasses classical mechanics as a special case of projection onto a single‑origin coordinate system:

1. When a single‑origin external Cartesian coordinate system is introduced, the MOC intrinsic geometric quantities can be projected onto classical coordinates, velocities, and accelerations.

2. Intrinsic curvature projects to gravitational mass, relative curvature projects to orbital distance, and torsion projects to angular momentum.

3. The MOC evolution equations can be strictly reduced to Newton’s three‑body equations of motion.

4. Classical chaos, non‑integrability, and unsolvability are all consequences of structural information loss and geometric distortion during the projection process.

Classical mechanics is a low‑dimensional projective representation of the MOC geometric system; the MOC system is the high‑dimensional essential origin of classical mechanics.

7. Conclusion

This paper has fully constructed the complete three‑body dynamic system under the MOC framework with curvature–torsion bidirectional coupling. It systematically formalizes the four core dynamic equations, presents a unified compact vector form, derives global closed conservation laws and intrinsic geometric invariants, and achieves a complete closed loop from the minimalist equilibrium model to the full‑dimensional realistic dynamics of the MOC three‑body theory.

The core conclusions of this paper are as follows:

1. The essence of realistic three‑body dynamics is the self‑consistent coupling evolution of intrinsic curvature, orbital relative curvature, and spin torsion, without any need for derivative concepts such as force, mass, or inertial frames.

2. The spin torsion evolution equation, orbital curvature constraint equation, intrinsic curvature dynamic equation, and global dual constraint equation formalized herein constitute the core mathematical skeleton of the MOC system.

3. Under the MOC framework, the three‑body system possesses global strict conservation laws and geometric invariants, with no intrinsic chaos, divergence, or non‑integrability.

4. The classical chaos and unsolvability of the three‑body problem are representational effects arising from projection onto a single‑origin coordinate system, not intrinsic properties of the system.

5. The unified master equation \mathcal{D} \star \mathcal{R} = 0 is mathematically isomorphic to the Yang–Mills gauge field equation, laying the fundamental foundation for subsequent generalization to a unified field theory.

This paper marks the completion of a complete, self‑consistent, ultimate dynamic description of the three‑body problem by the MOC multi‑origin curvature theory. Subsequent papers in the series will successively address: rigorous analytical solutions for symmetric configurations, a paradigmatic reconstruction of Yang–Mills field theory from the MOC perspective, a projective breakdown theory of chaos, generalization to N‑body systems, and the program of unified geometric dynamics.

Acknowledgments

This paper is dedicated to all rational explorers who break free from the obsession with coordinates and pursue the geometric origin of the physical world.