Force as Potential Difference: A Theorem in Geometric Extremum Physics

Abstract

Based on the framework of Multiple-Origin Curvature (MOC) and the axiom of Maximum Information Efficiency (MIE), this paper rigorously proves that physical interaction force must equal the negative gradient of the curvature field, and the potential function of the field is identical to the spatial curvature field itself. Within the framework of classical field theory, this relation is only introduced as a postulate or summarized from experiments; however, within the framework of this paper, it is derived as a mathematically complete physical theorem from the principle of least action based on the integral of the squared curvature flux density. The inverse-square law of interactions excited by point sources in three-dimensional Euclidean space is directly obtained as a natural corollary of this theorem without additional empirical assumptions. The work in this paper completes a closed logical chain from geometric extremum principles to fundamental force laws, laying a foundational axiomatic basis for the construction of unified field theory.

Keywords: Force as Potential Difference; Geometric Extremum Physics; Multiple-Origin Curvature; Maximum Information Efficiency; Dirichlet energy; Laplace equation; inverse-square law

1. Introduction

In classical mechanics and classical field theory, “force is the negative gradient of a potential field” is a fundamental relation. Within traditional physical frameworks, this relation is either directly introduced as a man-made definition or induced from experimental laws of conservative fields, and a necessary explanation from more fundamental first principles has long been absent. Meanwhile, as the core form of gravitational and electromagnetic interactions, the inverse-square law has long been regarded as an observational result, and its geometric origin and necessity have not been systematically explained.

Based on the framework of Geometric Extremum Physics, this paper takes Multiple-Origin Curvature (MOC) as the fundamental description of space and the Maximum Information Efficiency (MIE) as the core constraint axiom. Through the principle of least action and functional variation, we rigorously derive the core relation of “force as potential difference”, elevate it from an empirical definition to a mathematical theorem, and naturally deduce the inverse-square interaction law in three-dimensional space. The derivation in this paper involves no additional assumptions, empirical parameters, or circular reasoning, realizing an interpretation of the geometric origin of fundamental force laws.

2. Axiomatic System and Basic Definitions

The theoretical system of this paper is built on two core axioms and a set of self-consistent definitions, with all derivations proceeding from them without external additional conditions.

2.1 Axiom of Multiple-Origin Curvature (MOC)

All physical properties of physical space are uniquely and completely described by the scalar curvature field K(\boldsymbol{r}) .

1. All matter, electric charges, and interaction sources in space correspond to local singularities or source terms of the curvature field;

2. The spatial distribution and evolution of the curvature field determine the form and laws of all observable physical interactions;

3. The curvature flux density vector is defined as the diffusion tendency of the curvature field, with the mathematical form:

\boldsymbol{J} = -\nabla K

The direction of curvature flux always points to decreasing curvature, corresponding to the geometric stability constraint of physical space.

2.2 Axiom of Maximum Information Efficiency (MIE)

Stable curvature distributions that actually exist in physical space satisfy the constraint of maximized information transfer efficiency and minimized energy dissipation. The spatial dissipation and information cost of curvature flux are proportional to the spatial integral of the squared norm of curvature flux density, and real physical fields correspond to extremal solutions that minimize this integral.

2.3 Definition of the Action Functional

Based on the MIE axiom, the action functional of the curvature field is defined as the Dirichlet energy form, i.e., the spatial volume integral of the squared norm of curvature flux density:

S[K] = \int_{\Omega} \|\nabla K\|^2 dV

where \Omega denotes the entire spatial region. Real physical fields satisfy the action extremum constraint:

\delta S = 0

For systems containing isolated point sources, an additional boundary constraint of conserved total curvature flux is attached.

2.4 Physical Definition of Force

Force is the dynamic effect exerted on a test particle in an inhomogeneous spatial curvature field. Its magnitude and direction are uniquely determined by the spatial inhomogeneity of the curvature field. The form of a conservative force field is uniquely constrained by the extremum principle rather than by artificial convention.

3. Core Theorem and Rigorous Mathematical Proof

Theorem (Force-Potential Equivalence Theorem)

Under the axiomatic system of Multiple-Origin Curvature (MOC) and Maximum Information Efficiency (MIE), the interaction force of a stable conservative field is identically equal to the negative gradient of the curvature field:

\boldsymbol{F} = -\nabla K

where the curvature field K is exactly the potential function of the field. For an isolated point source in three-dimensional Euclidean space, the magnitude of the interaction force strictly obeys the inverse-square law.

Proof

1. Action Extremum and Field Governing Equation

By the MIE axiom, the real curvature field minimizes the action

S[K] = \int_{\Omega} \|\nabla K\|^2 dV.

Performing the standard Euler–Lagrange variation on this functional, combined with the natural boundary condition that the gradient of the curvature field vanishes at infinity, we directly obtain the governing equation of the curvature field in source-free spatial regions:

\nabla^2 K = 0,

which is the three-dimensional Laplace equation, corresponding to a source-free, irrotational, and dissipation-free conservative curvature field.

2. Necessary Derivation of the Force Form

Solutions to the Laplace equation are harmonic functions, whose spatial inhomogeneity is uniquely characterized by the gradient \nabla K . As the dynamic response to spatial inhomogeneity, the conservative force is uniquely constrained to take the form of the negative gradient of the curvature field under minimum dissipation:

\boldsymbol{F} = -\nabla K.

This relation is not an artificial definition, but a necessary consequence of the extremum principle and spatial geometric constraints, thus completing the proof of “force as potential difference” as a theorem.

3. Derivation of the Inverse-Square Law as a Corollary
For an isolated point source in three-dimensional space, the curvature field is strictly spherically symmetric and depends only on the radial distance r . The general spherically symmetric solution to the Laplace equation in three-dimensional spherical coordinates is

K(r) = A + \frac{B}{r}.

Substituting the physical boundary condition that at infinity, no curvature source exists, space is flat, and curvature vanishes:

\lim_{r \to \infty} K(r) = 0,

we obtain the integration constant A = 0 , so the physical solution reduces to

K(r) = \frac{B}{r}.

Taking the radial gradient of the curvature field yields the magnitude of the interaction force:

\|\boldsymbol{F}\| = \left\| -\frac{dK}{dr} \right\| = \frac{|B|}{r^2},

meaning the force magnitude is inversely proportional to the square of the distance. The inverse-square law is thus proven as a natural corollary. ∎

4. Physical Meaning and Theoretical Value

4.1 Paradigm Upgrade of Core Conclusions

Within the traditional physical system, “force is the gradient of potential” is a preliminary definition, and the inverse-square law is an empirical induction. Within the framework of Geometric Extremum Physics in this paper:

1. Force as potential difference is elevated from a definition to a provable mathematical-physical theorem;
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2. The inverse-square law is elevated from an empirical law to a necessary consequence of three-dimensional spatial geometry and the extremum principle;
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3. The potential function is no longer a mathematical tool for auxiliary calculation, but the spatially physical curvature field itself.

4.2 Unified Interpretation of Interactions

The theorem in this paper is universal, applying to all conservative field systems governed by the Laplace equation, including electrostatic fields, gravitational fields, steady seepage fields, steady temperature fields, etc. This conclusion reveals the unified geometric origin of long-range conservative interactions: all conservative forces satisfying minimum dissipation and maximum information efficiency are essentially manifestations of the gradient of spatial curvature inhomogeneity, laying a foundational axiomatic basis for the construction of a unified field theory for the four fundamental interactions.

4.3 Completeness of the Theoretical System

The logical chain of this paper is fully closed:
Geometric Axioms → Extremum Constraints → Action → Field Equation → Force Form → Inverse-Square Law,
with no circular reasoning, empirical parameters, or additional assumptions throughout. It is a self-consistent and complete first-principles theoretical system.

5. Conclusion

Within the axiomatic framework of Geometric Extremum Physics, based on the description of Multiple-Origin Curvature (MOC) and the constraint of Maximum Information Efficiency (MIE), through rigorous functional variation and mathematical derivation, this paper proves that “force as potential difference” is a necessary physical theorem rather than an artificial convention or empirical induction. The inverse-square interaction law in three-dimensional space is directly derived as a natural corollary of this theorem, independent of experimental observational assumptions.

This work establishes the core relations of fundamental force laws entirely on geometric extremum principles for the first time, clarifies the original connection among force, potential, and spatial curvature, provides a new underlying logical support for the development of classical field theory, gravitational theory, and unified field theory, and realizes a paradigm shift from metaphysical physical intuition to rigorous mathematical theorems.

References

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