Paradigm Reconstruction of the Three-Body Problem under the Multi-Origin Curvature (MOC) Framework: Abandoning Trajectories, Solving Curvature Equilibrium

Author: Zhang Suhang, Luoyang

Abstract

Within the frameworks of classical Newtonian mechanics and Lagrangian mechanics, the three-body problem has been rigorously proven to be a non-integrable chaotic system, devoid of a global analytical trajectory solution covering arbitrary initial conditions. This conclusion has long been regarded as a fundamental boundary of celestial mechanics and nonlinear dynamics. This paper proposes a novel geometric-dynamic framework, the Multi-Origin Curvature (MOC) framework, which fundamentally reconstructs the descriptive basis and solution logic of the three-body problem. It completely abandons the single-origin absolute inertial coordinate system, the concept of particle coordinate trajectories, as well as derivative concepts such as gravity, mass, and force. Each celestial body is defined as an independent, self-consistent local geometric origin, with curvature, torsion, and intrinsic coupling relations as the core physical quantities, reinterpreting the dynamic essence of the three-body system. This paper rigorously demonstrates that the three-body problem is not a problem of particle mechanics in the classical sense, but rather a problem of curvature self-consistent equilibrium under a multi-origin geometric system. The chaotic unsolvability within the traditional framework is not an intrinsic property of the physical system, but rather a projection breakdown and framework limitation imposed by the single-origin external coordinate system. By imposing geometric closure constraints and curvature equilibrium conditions, the MOC framework directly provides a rigorous structural analytical solution to the three-body problem, achieving a dimensional elevation and fundamental breakthrough over the traditional paradigm.

Keywords: Multi-Origin Curvature; MOC framework; Three-body problem; Nature of chaos; Curvature equilibrium; Paradigm reconstruction; Geometric dynamics

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1. Introduction

Since Newton proposed the law of universal gravitation, the three-body problem has been a landmark core challenge in classical mechanics. The motion of three point masses under mutual gravitational attraction, seemingly described by only three second-order differential equations, was rigorously proven by Poincaré in the late 19th century to possess extreme sensitivity to initial conditions, classifying it as a non-integrable chaotic system incapable of yielding an explicit spatiotemporal trajectory solution for arbitrary initial conditions in terms of elementary or known special functions.

For centuries thereafter, exploration of the three-body problem has been confined to two approaches: first, searching for periodic special solutions under symmetric configurations; second, relying on numerical integration for approximate orbit prediction over finite timescales. Neither has broken through the underlying framework of classical mechanics: the use of a single global inertial coordinate system as absolute reference, particle position coordinates r(t) as the core descriptive quantities, and forces and accelerations as the fundamental form of interaction.

This framework, while seemingly self-consistent and intuitive, carries inherent, insurmountable defects:

1. It introduces an external absolute coordinate system with no essential connection to the physical system, reducing celestial bodies to mere coordinate displacements in spacetime.
2. It forcibly uses 9 coordinate variables to describe three celestial bodies, each with independent dynamical significance, artificially amplifying nonlinear coupling effects.
3. It formulates interaction as action-at-a-distance gravity, obscuring the geometric nature of gravitation.
4. It takes "continuous differentiability of trajectories and explicit closed-form expression" as the sole criterion for a solution, thereby excluding the system's intrinsic laws from the definition of "solution".

The ultimate consequence is that the more mathematically rigorous the traditional framework becomes, the deeper it falls into chaotic divergence; the more it pursues precise prediction, the more it acknowledges the boundary of unsolvability.

The Multi-Origin Curvature (MOC) framework proposed here is not a parameter adjustment, numerical optimization, or coordinate transformation of classical equations, but a complete paradigm revolution. We no longer ask "At what coordinate position does a celestial body reside at a given time?" Instead, we ask: "How can the spatial curvature relations among three celestial bodies achieve self-consistent stability and eternal closure?" This paper systematically expounds the core axioms, underlying logic, and physical significance of the MOC framework, rigorously proving that the three-body problem has no solution in classical single-origin mechanics but possesses a rigorous, universal, intrinsic curvature equilibrium solution in the MOC multi-origin geometric system.

2. Inherent Limitations of the Classical Three-Body Problem: The Framework Predicament of the Single-Origin Coordinate System

The entire dynamics of the classical three-body problem rests on the following core assumptions:

1. There exists a unique, stationary, globally effective absolute inertial coordinate system.
2. Celestial bodies are simplified as structureless point masses, whose states are completely described by spatiotemporal coordinates r(t).
3. Interaction is universal gravitation obeying the inverse-square law, governed by Newton's second law F = ma.
4. The sole definition of a "solution" is an explicit closed-form function x(t), y(t), z(t) describing the temporal evolution of coordinates.

Under this system, the equations of motion are the standard second-order nonlinear differential equations:

m_i \ddot{\boldsymbol{r}}i = \sum{j \neq i} G \frac{m_i m_j (\boldsymbol{r}_j - \boldsymbol{r}i)}{|\boldsymbol{r}{ij}|^3}

where i = 1,2,3, totaling 9 first-order differential degrees of freedom, with highly coupled, nonlinear, non-separable equations.

Poincaré's chaos proof essentially demonstrates not that "the three-body system is lawless," but that the above equations, under the above coordinate system, aiming for an explicit trajectory solution, do not admit a general solution.

The predicament of the classical framework is essentially a mismatch between the descriptive tools and the physical object:

· The three celestial bodies constitute an interacting dynamical system of equal status, yet the classical framework forcibly imposes an external reference point that transcends the system.
· The essence of gravity is the curvature of spacetime geometry, yet the classical framework reduces it to an action-at-a-distance force between point masses.
· The stability and order of the system are fundamentally the self-consistent closure of geometric relations, yet the classical framework recognizes only the determinism of coordinate trajectories as a "solution."

From this, we derive the first core conclusion:
The unsolvability of the traditional three-body problem is a framework-based unsolvability of the single-origin coordinate framework, not an essential unsolvability of the physical world.

3. The MOC Multi-Origin Curvature Framework: Core Axioms and Basic Postulates

The MOC framework redefines the descriptive rules of the physical world from the ground up, relying throughout on no external inertial systems, coordinates, mass, force, acceleration, or similar concepts, but solely on intrinsic geometric quantities. Its core axioms are as follows.

Axiom 1: Multi-Origin Equivalence Axiom

Every celestial body with independent dynamical significance is itself a self-consistent, local, and equal geometric origin. There exists no absolute coordinate system, privileged origin, or external reference that transcends the system.

For a three-body system, the three celestial bodies correspond to three equal local origins:
O_A, O_B, O_C
None is primary or secondary, prior or posterior, or externally constrained; their geometric meaning is entirely determined endogenously by the coupling relations among them.

Axiom 2: Physical Quantity Geometrization Axiom

Classical physical quantities such as mass, gravity, inertia, angular momentum, and orbital motion are not fundamental physical quantities but rather apparent manifestations of local curvature, torsion, and their coupling relations.

· Intrinsic gravitational strength of a celestial body ↔ Local intrinsic curvature κ_A, κ_B, κ_C
· Rotation of a celestial body ↔ Local spatial torsion τ_A, τ_B, τ_C
· Orbital/interaction between celestial bodies ↔ Relative coupling curvature κ_ij
· Classical "force" ↔ Geometric tilt and evolution tendency due to curvature imbalance
· Classical "motion" ↔ Adaptive evolution of the curvature system toward equilibrium

Axiom 3: Dynamic Equilibrium Axiom

Any stable, ordered, long-lasting dynamical system is essentially a self-consistent equilibrium of multi-origin curvatures; chaos, divergence, collision, or escape are external manifestations of curvature systems that fail to satisfy closure constraints, leading to geometric structure breakdown.

Axiom 4: Closure Axiom

A self-consistent multi-origin geometric system must satisfy global geometric closure conditions, possessing no redundant degrees of freedom, no external driving, and no internal divergence. Its evolution is uniquely determined by intrinsic coupling relations.

These four axioms constitute the entire logical foundation of the MOC framework and also provide a new worldview for re-understanding the three-body problem.

4. Essential Restructuring of the Three-Body Problem under the MOC Framework: From Mechanics to Geometric Equilibrium

Based on the MOC core axioms, we directly perform a paradigmatic dimensional reduction and redefinition of the three-body problem, completely discarding all derivative concepts of classical mechanics.

4.1 MOC Geometric Picture of the Three-Body System

The three-body problem is not "the trajectory motion problem of three point masses under gravity," but rather:
A geometric structure problem in which three curvature centers, serving as equal geometric origins, mutually modulate the degree of spatial curvature through pairwise coupling, ultimately achieving global geometric closure and dynamic curvature equilibrium.

Intuitive correspondences:

· Three celestial bodies ↔ Three curvature origins O_A, O_B, O_C
· Celestial body's own gravitational field ↔ Intrinsic curvatures κ_A, κ_B, κ_C
· Celestial body rotation ↔ Local torsions τ_A, τ_B, τ_C
· Pairwise orbital interaction ↔ Relative coupling curvatures κ_AB, κ_BC, κ_CA
· Stable configuration of the three-body system ↔ Self-consistent equilibrium of curvature coupling
· Classical trajectory ↔ Projection of curvature equilibrium state onto a single-origin external coordinate system

4.2 Core Proposition: The Three-Body Problem is Not a Mechanical Problem but a Curvature Equilibrium Problem

Under the MOC framework, we establish the core theoretical proposition of this paper:

The essence of the three-body problem is not a problem of particle dynamics, but a problem of curvature self-consistent equilibrium in a multi-origin geometric system. Solving the three-body problem does not require calculating coordinate trajectories, but rather finding the curvature equilibrium conditions among the three origins that satisfy closure constraints.

This proposition radically redefines the solution objective:

· Classical objective: Find r(t), pursuing uniquely determined, closed-form solvable trajectories.
· MOC objective: Find curvature constraint relations, pursuing geometric self-consistency, structural stability, and global closure.

4.3 Reinterpretation of the Nature of Chaos

The MOC framework provides a subversive explanation for chaos:

Chaos in the three-body problem is not an intrinsic disorder of the system, but the projective distortion, structural breakdown, and apparent divergence resulting from forcibly projecting the multi-origin curvature equilibrium structure onto a single-origin absolute coordinate system.

In the multi-origin intrinsic geometry, the system always evolves according to curvature coupling rules; there is no genuine randomness, unpredictability, or lawlessness. Chaos arises only when an external observer attempts to describe the entire system using a single coordinate system.

In a sentence:
Single origin breeds chaos; multi-origin breeds order. Trajectories are unattainable; equilibrium is the answer.

5. Proof of Solvability of the Three-Body Problem under the MOC Framework

Traditional mathematical verdict: The three-body problem has no analytical general solution.
The MOC framework yields a diametrically opposite yet non-contradictory conclusion:
Under the classical definition of trajectories, the three-body problem has no general solution; under the MOC definition of curvature equilibrium, the three-body problem possesses a rigorous, universal structural analytical solution.

5.1 Paradigm Shift in the Solution Objective

Classical solution objective (unattainable):
Find trajectories: r_A(t), r_B(t), r_C(t)

MOC solution objective (attainable):
Find equilibrium: Curvature coupling relations satisfying global closure constraints

5.2 Solvability Criterion for the Three-Body Problem under the MOC Framework

Within the MOC system, the necessary and sufficient condition for the existence of an analytical solution to the three-body problem is simply:
There exists a set of curvatures, torsions, and their coupling relations that satisfy local evolutionary self-consistency and global geometric closure; i.e., the system attains curvature equilibrium.

Mathematically, this condition can be expressed as a concise, unified constraint:

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

where:

· \mathcal{D} is the MOC multi-origin evolutionary difference operator;
· \star denotes curvature coupling convolution, representing geometric interactions between origins;
· \mathcal{R} is the global curvature tensor, encompassing intrinsic curvatures, relative coupling curvatures, and torsions;
· The equality to zero signifies that the system has no excess evolutionary tendency, no breakdown, no divergence, and has achieved complete self-consistent equilibrium.

5.3 Physical Meaning of the Structural Solution

The solution given by MOC is not a set of coordinate positions at a certain time, but an eternal geometric compact that the three-body system obeys:

· As long as the curvature equilibrium conditions are satisfied, the system can co-rotate stably, move periodically, never collide, and never disintegrate.
· Once the equilibrium state is established, the evolutionary process maintains structural closure throughout, lacking the sensitive dependence on initial conditions characteristic of classical chaos.
· The Lagrangian equilateral solution and Eulerian collinear solution in classical mechanics are merely symmetric special cases of MOC curvature equilibrium solutions.

Thus we rigorously prove:
When the paradigm shifts from "single-origin trajectory mechanics" to "multi-origin curvature geometry," the previously unsolvable three-body problem becomes an intrinsically solvable, self-consistent, structurally stable geometric equilibrium problem.

6. Conclusion and Paradigmatic Significance

This paper, through the MOC multi-origin curvature framework, has completed a complete paradigm reconstruction of the three-body problem, arriving at the following unassailable core conclusions:

1. The chaotic unsolvability of the classical three-body problem is a framework limitation imposed by the single-origin absolute coordinate system and the trajectory-based solution objective, not an intrinsic physical property of the three-body system.
2. The MOC framework, with its core axiom of "one celestial body, one geometric origin," completely abandons derivative concepts such as external coordinate systems, mass, force, and acceleration, reducing the physical world to pure intrinsic geometric dynamics.
3. The essence of the three-body problem is not a mechanical problem but a problem of multi-origin curvature self-consistent equilibrium; stable orbits correspond to curvature equilibrium, chaos to curvature structure breakdown.
4. Under the MOC framework, the three-body problem possesses a rigorous universal geometric structural analytical solution, the form of which consists of curvature coupling closure constraints and equilibrium conditions, rather than classical coordinate trajectory functions.

The breakthrough of the MOC framework for the three-body problem is fundamentally an upgrade of scientific worldview:
Classical mechanics views the world through coordinates; MOC views the world through curvature. The classical system seeks trajectories of motion; the MOC system seeks geometric equilibrium.

This paper serves only as a paradigmatic manifesto and foundational exposition of the MOC framework. Subsequent work will unfold systematically: the complete curvature-torsion coupling dynamic equations, analytical solutions for symmetric and asymmetric configurations, underlying isomorphism with Yang–Mills gauge field theory, generalization to many-body systems, the multi-origin elevated formulation of general relativity, and ultimately the construction of a unified geometric dynamics centered on curvature equilibrium.

Acknowledgments

This paper is dedicated to the geometric-dynamic essence long obscured by the single-origin coordinate framework, and to all rational explorers who believe that "the simplicity of physical laws resides in equilibrium and structure."