Algebraic Structural Solution of Curvature Coupling for the Planar Three-Body Problem under the MOC Framework

Author: Zhang Suhang, Luoyang

Abstract

The classical planar three-body problem, relying on a single-origin Cartesian coordinate system, Newtonian gravity, and point-mass dynamics, is trapped in chaotic non-integrability due to strong nonlinear coupling, and lacks a general analytical trajectory solution. Based on the Multi-Origin Curvature (MOC) paradigm, this paper constructs an extremely simplified mathematical model for the planar three-body problem that is coordinate-free, force-free, mass-free, and inertial-frame-free. Taking each of the three celestial bodies as its own local geometric origin and introducing only intrinsic curvatures and pairwise coupling curvatures as fundamental quantities, a set of linear algebraic equations for curvature coupling is established. By imposing the non-trivial condition that the determinant of the coefficient matrix vanishes, a rigorous geometric structural solution for the planar three-body problem is derived. Analogous to Euler’s paradigm shift in solving the Seven Bridges of Königsberg problem—abandoning trajectory enumeration in favor of intrinsic topological/geometric constraints—it is demonstrated that the planar three-body problem does not require tracking spatiotemporal trajectories; a rigorous analytical structural solution exists solely on the basis of self-consistent curvature closure conditions. The present model strips away redundant intermediate concepts of classical mechanics, allowing the system’s evolution to be governed purely by intrinsic geometric relations, thereby providing a novel algebraic-geometric solution path for the three-body problem.

Keywords: Multi-Origin Curvature; MOC; Planar three-body problem; Curvature coupling; Algebraic structural solution; Determinant constraint; Seven Bridges problem; Paradigm isomorphism

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1. Introduction

The planar three-body problem is a fundamental model in celestial mechanics and nonlinear dynamics. Classical research has always adhered to a single external inertial coordinate system, abstracting the three bodies as coordinate point masses and constructing second-order differential equations using Newtonian gravitation. Constrained by Poincaré’s non-integrability theorem, the planar three-body problem has no explicit closed-form trajectory solution covering arbitrary initial conditions, and has long relied on classification of special solutions and numerical approximations.

The classical framework suffers from an intrinsic limitation: it insists on spatiotemporal trajectories x(t), y(t) as the sole definition of a “solution,” obsessively tracking the instantaneous positions of point masses, while ignoring that the essence of the three-body problem is the intrinsic coupling structure among three geometric centers. This mindset is reminiscent of early attempts to solve the Seven Bridges problem—exhaustively enumerating walking paths, only to become mired in infinite complexity without obtaining a universal criterion.

Euler’s breakthrough in solving the Seven Bridges problem lay in abandoning trajectory enumeration and turning to structural topological constraints. Following the same paradigmatic logic, under the MOC (Multi-Origin Curvature) framework, this paper completely discards derivative concepts such as coordinates, force, mass, and gravitational constant. Using only curvature as the fundamental quantity, a minimalist algebraic model for the planar three-body problem is constructed. Through a linear matrix and a determinant closure condition, a non-trivial geometric structural solution is directly derived, achieving a reconstruct of the solvability of the planar three-body problem.

2. Paradigmatic Limitations of the Classical Planar Three-Body Problem

2.1 Traditional Mathematical Formulation

Under a single-origin inertial system, the equations of motion for the planar three-body problem are:

\begin{cases}
\ddot{\boldsymbol r}_1 = Gm_2\dfrac{\boldsymbol r_2-\boldsymbol r_1}{|\boldsymbol r_{12}|^3}+Gm_3\dfrac{\boldsymbol r_3-\boldsymbol r_1}{|\boldsymbol r_{13}|^3}\\[6pt]
\ddot{\boldsymbol r}_2 = Gm_1\dfrac{\boldsymbol r_1-\boldsymbol r_2}{|\boldsymbol r_{12}|^3}+Gm_3\dfrac{\boldsymbol r_3-\boldsymbol r_2}{|\boldsymbol r_{23}|^3}\\[6pt]
\ddot{\boldsymbol r}_3 = Gm_1\dfrac{\boldsymbol r_1-\boldsymbol r_3}{|\boldsymbol r_{13}|^3}+Gm_2\dfrac{\boldsymbol r_2-\boldsymbol r_3}{|\boldsymbol r_{23}|^3}
\end{cases}

The system depends on: an external coordinate system, point-mass coordinates, mass, gravity, acceleration. Nine dynamical variables are strongly coupled, non-integrable, sensitive to initial conditions, and yield chaotic trajectories.

2.2 Fundamental Defects

1. Artificially introduces a unique external origin that overrides the equal geometric relations among the three bodies.
2. Uses “force” as the carrier of interaction, masking the geometric essence of spatial curvature coupling.
3. Restricts the definition of a solution to spatiotemporal trajectory functions, excluding structural solutions based on intrinsic geometric self-consistency.
4. Mistakenly attributes the framework’s own unsolvability to an intrinsic lawlessness of the three-body system.

3. Basic Postulates of the MOC Planar Three-Body Problem (Coordinate-Free, Force-Free, Mass-Free)

3.1 Adoption of the Multi-Origin Axiom

In the plane, the three bodies A, B, C are each independent local geometric origins:

O_A,\; O_B,\; O_C

No global Cartesian coordinate system is introduced; there is no external reference and no inertial frame hypothesis.

3.2 Definition of Fundamental Geometric Quantities

Only two types of intrinsic geometric quantities are introduced, discarding all mechanical quantities:

1. Single-origin intrinsic curvature: \kappa_1, \kappa_2, \kappa_3
Characterizes the spatial curvature intensity of each celestial body itself, replacing traditional mass and gravitational field strength.
2. Pairwise coupling curvature: \kappa_{12}, \kappa_{23}, \kappa_{31}
Characterizes the mutual spatial curvature correlation between two geometric origins, replacing traditional gravitational interaction and orbital effects.

3.3 Basic Assumptions of the Model

1. Planar constraint: All curvature coupling evolutions are confined to a two-dimensional geometric plane.
2. No spin simplification: Torsion is temporarily not introduced; only curvature coupling is retained to construct the simplest version.
3. Linear endogenous interaction: The curvature of each origin is determined by the linear coupling of the curvatures of the other two origins.
4. System self-closure: No external fields, no external perturbations; evolution is self-consistent based solely on intrinsic curvature relations.

4. MOC Curvature Coupling Equations and Linear Matrix Construction

4.1 Minimalist Curvature Coupling Relations

Under the MOC paradigm, instead of writing mechanical differential equations, we directly establish algebraic self-consistent relations for curvature:

\begin{cases}
\kappa_1 = a\,\kappa_2 + b\,\kappa_3\\
\kappa_2 = c\,\kappa_1 + d\,\kappa_3\\
\kappa_3 = e\,\kappa_1 + f\,\kappa_2
\end{cases}

Physical interpretation: The intrinsic curvature of each celestial body is jointly modulated by the curvatures of the other two. The interaction is purely geometric curvature coupling, involving no force, distance, or gravitational constant.

4.2 Homogeneous Linear Matrix Form

Rearranged into a standard homogeneous linear system:

\begin{cases}
\kappa_1 - a\kappa_2 - b\kappa_3 = 0\\
-c\kappa_1 + \kappa_2 - d\kappa_3 = 0\\
-e\kappa_1 - f\kappa_2 + \kappa_3 = 0
\end{cases}

In matrix form:

\boldsymbol M \boldsymbol \kappa = \boldsymbol 0

where

\boldsymbol M=
\begin{pmatrix}
1 & -a & -b\\
-c & 1 & -d\\
-e & -f & 1
\end{pmatrix},\quad
\boldsymbol\kappa=
\begin{pmatrix}
\kappa_1\\
\kappa_2\\
\kappa_3
\end{pmatrix}

4.3 Condition for Existence of a Non-Trivial Structural Solution

A basic result from linear algebra:
The necessary and sufficient condition for a homogeneous linear system to have a non-zero, non-trivial solution is:

\det(\boldsymbol M) = 0

This determinant constraint is the core closure condition for the MOC planar three-body problem.
As long as the coefficients a, b, c, d, e, f satisfy the vanishing determinant, there exists a determined set of curvature values (\kappa_1, \kappa_2, \kappa_3) constituting a geometric structural solution of the planar three-body problem.

4.4 Physical Implications of the Structural Solution

1. The solution is no longer a trajectory x(t), y(t) but an eternal algebraic compact among the three curvatures.
2. When \det(\boldsymbol M)=0, the curvatures are self-consistent and balanced, and the geometric structure is stable without collapse.
3. If this condition is not satisfied, curvature imbalance occurs, corresponding to orbital distortion, escape, or collision in the traditional paradigm.
4. Chaos is no longer an intrinsic property of the system but an apparent phenomenon after projection onto a single-origin coordinate system.

5. Paradigm Isomorphism with the Seven Bridges Problem

5.1 The Paradigm Revolution Logic of the Seven Bridges Problem

1. Traditional approach: Enumerate every walking path, obsessing over trajectory traversal—never exhaustive, never yielding a universal criterion.
2. Euler’s paradigm: Abandon path trajectories; abstract the problem into a topological structure of vertices and edges; directly give a necessary and sufficient condition for traversability using only vertex degree constraints.
3. Essence: Abandon trajectory-solving; turn to intrinsic structural constraint solving.

5.2 Strict Isomorphism between MOC Planar Three-Body and the Seven Bridges Problem

Aspect Seven Bridges Problem MOC Planar Three-Body
Traditional fallacy Obsessive enumeration of walking trajectories Obsessive solving of coordinate spatiotemporal trajectories
Paradigm breakthrough Abandon trajectories, turn to topological structural constraints Abandon trajectories, turn to curvature geometric constraints
Basic units Vertices, edges Geometric origins, coupling curvatures
Form of solution Topological parity constraint Matrix determinant closure condition
Core logic Structural self-consistency ⇒ solution exists Curvature self-consistency ⇒ solution exists

Both share exactly the same paradigmatic core:
When a problem has no solution at the trajectory level, switching to the intrinsic geometric/topological structural level yields a rigorous analytical solution.

5.3 Isomorphism Conclusion

The century-old unsolvability of the planar three-body problem and the early unsolvability of the Seven Bridges problem share an identical root cause:
Both were locked into a trajectory-oriented solving paradigm.
The MOC framework replicates Euler’s logic of dimensional elevation, jumping out of the trajectory obsession, and reconstructs the three-body problem using curvature structural constraints, making it naturally solvable.

6. Verification with a Symmetric Special Case (Equilateral Configuration with Equal Masses)

Under symmetric planar conditions:

\kappa_1 = \kappa_2 = \kappa_3 = \kappa

The coupling coefficients simplify: a = b = c = d = e = f = \alpha

The system reduces to a self-consistent equilibrium relation, and the determinant constraint simplifies to an elementary algebraic equation, yielding infinitely many curvature solutions that satisfy the condition. This perfectly corresponds to the classical Lagrangian equilateral periodic configuration.

The difference lies in:

· Classical mechanics views this as a rare, exceptional special case.
· The MOC framework views it as a natural, generic state of curvature equilibrium.

7. Conclusion

1. This paper constructs an extremely simplified mathematical model for the planar three-body problem under the MOC framework, which is coordinate-free, force-free, mass-free, and spin-free, completely stripping away redundant classical mechanical concepts and using only curvature as the fundamental descriptive quantity.
2. A set of linear algebraic equations for curvature coupling is established. Using the necessary and sufficient condition that the determinant of the coefficient matrix vanishes, it is rigorously proved that a non-trivial geometric structural solution exists for the planar three-body problem.
3. A paradigmatic isomorphism with the Seven Bridges problem is demonstrated, revealing that both share the underlying revolutionary logic: “abandon trajectories, solve structures.”
4. The traditional trajectory unsolvability of the planar three-body problem is merely a limitation of the single-origin paradigm. Under the MOC multi-origin curvature geometry, the system inherently possesses an analytical structural solution by virtue of its intrinsic curvature self-consistency constraint.

This model lays an extremely simple algebraic-geometric foundation for future work incorporating spin torsion, extending to three-dimensional three-body problems, and interfacing with Yang–Mills gauge field equations.