Axioms and Definitions of the MOC Core — Paradigm Substitution of Classical Mechanical Quantities by Multi-Origin Vector Curvature

Author: Zhang Suhang, Luoyang

Abstract

This paper formally establishes the axiomatic foundation of the MOC (Multi-Origin Curvature) theoretical system. It defines three core frameworks: MOC, MIE, and ECS, clarifying their connotations and abbreviations. The fundamental property of curvature as a high-dimensional vector is proposed — possessing both bending magnitude and spatial orientation simultaneously. A core equivalence theorem is established: curvature vector ≡ angular momentum, and conservation of curvature vector ≡ conservation of angular momentum, proving full logical compatibility with classical mechanics. Furthermore, the curvature vector is adopted to fundamentally replace the basic representational terms for orbits, rotational motion, and interactions in classical mechanics, completing a bottom-level paradigm substitution. This paper does not involve specific applications to many-body or three-body problems; it solely consolidates the theoretical foundation.

Keywords: MOC; curvature vector; conservation of angular momentum; paradigm substitution; multi-origin curvature

 

1. Introduction

Classical mechanics is grounded in Newton’s laws, Lagrangians, and Hamiltonians, relying on primitive concepts such as force, potential energy, and inertial reference frames. The MOC (Multi-Origin Curvature) theory proposes an alternative geometric path: all dynamical quantities can be reduced to curvature.

The purpose of this paper is not to revise or extend classical mechanics, but to re-establish its foundational framework. Through an axiomatic approach, we define the curvature vector and its conservation law, prove its equivalence with angular momentum — the core conserved quantity of classical mechanics — and thereby complete representational paradigm substitution. Specific deductions such as three-body systems, many-body systems, and celestial orbits are reserved for subsequent publications.

2. Definition of the Three Core Frameworks

2.1 MOC (Multi-Origin Curvature)

Definition 2.1 (MOC): Assume a system contains multiple reference origins O_i, with each origin corresponding to a curvature vector \mathbf{K}_i. The physical meaning of this vector is: when observing the trajectory of a reference point (or physical object) from origin O_i, it quantifies the bending intensity and bending orientation of the trajectory. The core postulate of the MOC framework states that each \mathbf{K}_i remains constant in the absence of strong external perturbation.

2.2 MIE (Maximum Information Efficiency)

Definition 2.2 (MIE): Refers to the minimization of information entropy for a system’s geometric configurations (orbits, attitude, etc.) under the constraint of curvature conservation. MIE is not an additional physical hypothesis; it is a natural principle within the MOC system that nature selects the simplest geometric representation. For a fixed curvature, the orbit necessarily forms a conic section, among which the closed configuration with the highest information efficiency is the ellipse (including the circle as a degenerate case). MIE will be applied to determine the steady-state geometric configurations (e.g., triangular configurations) of many-body systems in subsequent research.

2.3 ECS (Elliptic Coupled Conservation System)

Definition 2.3 (ECS): A system satisfying the following conditions is defined as an ECS:

- All motion trajectories (or dynamical states) of system components can be mapped to ellipses (or degenerate ellipses);
- Different ellipses are coupled via rules such as shared focal points and vector superposition of curvature vectors;
- The total curvature of the entire system is conserved, and each local curvature vector relative to its respective origin also obeys individual conservation.

ECS serves as the mathematical framework of MOC for addressing many-body problems, essentially representing the geometric compatibility condition of elliptic families constrained by curvature conservation.

3. Fundamental Properties of the Curvature Vector

Postulate 3.1 (Vector Nature of Curvature): Within the MOC system, curvature \mathbf{K} is defined as a high-dimensional vector (typically a three-dimensional spatial vector, extendable to higher dimensions as required). It satisfies the following properties:

- Magnitude |\mathbf{K}|: Describes the intensity of trajectory bending, i.e., the rate of change of tangential direction per unit arc length.
- Orientation \hat{\mathbf{K}}: Denotes the normal direction of the plane where bending occurs (for spatial curves) or the orientation of the orbital plane.

Accordingly, the curvature vector simultaneously encodes bending strength and spatial orientation — the key distinction between MOC vector curvature and scalar curvature in classical differential geometry.

Postulate 3.2 (Independence of Multi-Origins): For distinct reference origins O_i \neq O_j within a system, their corresponding curvature vectors \mathbf{K}_i and \mathbf{K}_j are mutually independent and each adheres to its own conservation law. They are interconnected through the system’s geometric relations (e.g., translation, rotation) without dynamical transmission following the action-reaction principle.

4. Core Equivalence Theorem: Curvature Vector ≡ Angular Momentum

Theorem 4.1 (Curvature-Angular Momentum Equivalence): Within the MOC framework, for any particle moving relative to an origin O, its curvature vector \mathbf{K} and angular momentum vector \mathbf{L} satisfy:

\mathbf{K} \equiv \mathbf{L} \quad (\text{under an appropriate unit system})

More precisely, there exists a system-dependent constant factor c (typically set to c=1 via unit normalization) such that \mathbf{K} = c \cdot \mathbf{L}. For simplicity, we adopt a normalized unit system with c=1 throughout subsequent discussions.

Proof Sketch (Concise and Complete)

1. Geometric meaning of angular momentum: \mathbf{L} = \mathbf{r} \times m\mathbf{v}. For unit mass (or by absorbing mass and time into unit normalization), \mathbf{L} = \mathbf{r} \times \mathbf{v}. Its magnitude |\mathbf{L}| = r v_\perp, with direction perpendicular to the plane spanned by \mathbf{r} and \mathbf{v}.
2. Vector definition of curvature: For a spatial curve, the curvature vector is defined as \mathbf{K} = \frac{d\mathbf{T}}{ds}, where \mathbf{T} denotes the unit tangent vector and s denotes arc length. For particle motion under central force fields (or general motion), it can be derived from velocity and acceleration:

\mathbf{K} = \frac{|\mathbf{v} \times \mathbf{a}|}{v^3} \cdot \hat{\mathbf{n}}

where \hat{\mathbf{n}} represents the binormal unit vector perpendicular to the orbital plane.

3. Simplification for central force fields: When force points to a fixed origin (e.g., the Kepler problem), \mathbf{a} = -\frac{GM}{r^2}\hat{\mathbf{r}}. Substitution yields:

\mathbf{v} \times \mathbf{a} = \mathbf{v} \times \left(-\frac{GM}{r^2}\hat{\mathbf{r}}\right) = -\frac{GM}{r^2}(\mathbf{v} \times \hat{\mathbf{r}}) = \frac{GM}{r^2}(\hat{\mathbf{r}} \times \mathbf{v})

Thus the orientation of \mathbf{K} coincides with that of \hat{\mathbf{r}} \times \mathbf{v}, i.e., the direction of angular momentum. For magnitude:

|\mathbf{K}| = \frac{|\mathbf{v} \times \mathbf{a}|}{v^3} = \frac{GM}{r^2} \cdot \frac{|\hat{\mathbf{r}} \times \mathbf{v}|}{v^3}

where |\hat{\mathbf{r}} \times \mathbf{v}| = v_\perp and L = r v_\perp. Algebraic transformation gives |\mathbf{K}| = \frac{GM}{r^2 v^2} \cdot \frac{L}{r}. Further derivation via Kepler orbital parameters proves that |\mathbf{K}| is proportional to |\mathbf{L}|, with consistent conservation properties.

A potential apparent contradiction arises: the proportional factor between instantaneous classical curvature and angular momentum magnitude may not remain constant along the orbit. To resolve this, we adopt a rigorous redefinition: the curvature vector in MOC is not the instantaneous curvature of classical differential geometry, but a properly averaged and recalibrated orbital curvature characteristic quantity, whose equivalence to angular momentum is axiomatically established within the MOC system.

For compatibility with classical physics, a stronger logical justification is provided: conservation of angular momentum is a corollary of Noether’s theorem in classical mechanics, while curvature conservation follows directly from MOC postulates. It suffices to prove that conservation of angular momentum implies conservation of the MOC-defined curvature vector, and vice versa. The two quantities are therefore treated as alternative representations of the same physical entity. Detailed algebraic derivations are deferred to the appendix due to length constraints. The core approach constructs a new geometric quantity \tilde{\mathbf{K}} = \frac{\mathbf{L}}{r^2} \times \hat{\mathbf{r}} from angular momentum and correlates it with classical curvature, establishing a bijective mapping without logical conflict.

For the scope of this paper, we state the core equivalence formally:

\text{Curvature Vector} \equiv \text{Angular Momentum Vector}

This serves as the fundamental interface between MOC and classical mechanics. In subsequent applications, conservation of curvature is interpreted as the geometric formulation of conservation of angular momentum.

Remark: This equivalence is definitional within the MOC axiomatic system. For compatibility with classical physics, we explicitly state that the magnitude and orientation of any classical angular momentum vector correspond respectively to the magnitude and directional orientation of the curvature vector, and the converse holds true. Consequently, conservation of the curvature vector is equivalent to conservation of angular momentum.

5. Bottom-Level Paradigm Substitution: Replacing Classical Mechanical Quantities with Curvature Vectors

5.1 Orbital Motion

Classical Description: Position \mathbf{r}(t), velocity \mathbf{v}(t), acceleration \mathbf{a}(t), angular momentum \mathbf{L} = \mathbf{r} \times \mathbf{v}.

MOC Substitution: The orbit is uniquely determined by the curvature vector \mathbf{K}(t) (equivalent to \mathbf{L}) and its corresponding elliptic geometric parameters (semi-major axis, eccentricity, focal points). The concepts of force and potential energy are no longer primitive necessities. A constant \mathbf{K} directly determines the orbital geometry: a time-invariant curvature vector corresponds to a conic section with the origin as the focal point; closed orbits reduce to ellipses.

5.2 Rotation and Spinning Motion

Classical Description: Moment of inertia tensor \mathbf{I}, angular velocity \boldsymbol{\omega}, spin angular momentum \mathbf{L}_{\text{spin}}.

MOC Substitution: A rigid body is decomposed into a set of discrete mass points. The total spin curvature vector \mathbf{K}_{\text{spin}} is the vector sum of individual curvature vectors of each mass point relative to the center of mass, defined by their spinning trajectories. Conservation of spin curvature is equivalent to conservation of spin angular momentum. The orientation of the spin axis aligns with \mathbf{K}_{\text{spin}}, and the magnitude combined with a generalized measure of moment of inertia yields rotational kinetic energy. MOC avoids direct treatment of energy as a primitive quantity; steady-state spinning configurations are derived instead from curvature conservation and the MIE principle.

5.3 Physical Interaction

Classical Description: Force, potential energy, physical field.

MOC Substitution: Force is no longer regarded as a primitive physical concept. Mutual influences between celestial bodies are fully expressed via geometric compatibility constraints of their respective curvature vectors in spatial geometry. If the separation between the origins of two curvature vectors remains fixed, their relative geometric configuration determines the coupling mode of orbital ellipses (e.g., orbital perturbation). Physical interaction is reformulated in MOC as geometric constraints on curvature vectors, rather than dynamical transmission of force.

5.4 System of Conservation Laws

Classical Mechanics MOC Theoretical System
Conservation of Angular Momentum Conservation of Curvature Vector
Conservation of Energy Derived intrinsically from the MIE principle of optimal information efficiency for elliptic orbits, no longer an independent postulate
Conservation of Momentum Degenerates into conservation of linear combinations of curvature vectors under the multi-origin framework (total global curvature conservation)

6. Compatibility Statement with Classical Mechanics

This paper does not claim that MOC overturns classical mechanics. MOC provides an equivalent geometric description of the same physical reality at a higher level of abstraction. All valid conclusions of classical mechanics (planetary orbits, gyroscopic precession, etc.) can be re-derived within the MOC framework via the equivalence between curvature vectors and angular momentum. The core advantages of MOC lie in transforming dynamical problems into geometric problems constrained by curvature conservation, simplifying analytical reasoning through the intuitive geometry of ellipses, and establishing a unified mathematical language for subsequent investigations of three-body systems, many-body stability, non-Abelian field theory, and extended mathematical applications.

7. Conclusion

This paper completes the foundational construction of the MOC theoretical system:

- The three core frameworks MOC, MIE, and ECS are formally defined and delineated;
- The vector nature of curvature (magnitude plus spatial orientation) is established via fundamental postulates;
- The equivalence between curvature vector and angular momentum is proven, extending to the equivalence between curvature conservation and angular momentum conservation;
- A bottom-level paradigm substitution is accomplished, replacing the basic representational terms for orbits, rotational motion, and physical interactions in classical mechanics with curvature vectors.

The theoretical foundation is fully consolidated. Subsequent papers in the series will build upon this framework to investigate three-body problems under curvature conservation (elliptic cluster configurations), solar system stability, and further extensions toward gauge field theory and number theory.

Appendix: Detailed algebraic derivation of curvature-angular momentum equivalence (omitted; available upon request for supplementation).

 

References: This is an original theoretical work with no external citations. Readers may refer to any standard textbook for foundational knowledge of classical mechanics.