Since the beginning of systematic observations of natural motions, “force” has always been the core concept used to explain all changes in the world. From the movement of celestial bodies to the falling of objects, from electromagnetic induction to particle interactions, whenever a state of motion changes, it is mostly attributed to the action of force. After centuries of development, the system of mechanics has become increasingly complete: Newton laid down the fundamental laws of motion for point particles; Lagrange and Hamilton unified conservative systems using potential functions and the principle of least action; Maxwell described electromagnetic interactions with field theory; Einstein reduced gravity to spacetime geometry; and gauge field theory further incorporated fundamental interactions into a framework of local symmetries. Within the cognitive boundaries of their respective eras, physicists have provided locally valid mathematical descriptions and computable models of force.

Nevertheless, although the potential function occupies a central position in analytical mechanics and field theory, the ultimate question — “What exactly is force?” — still lacks a unified answer rooted in the first principles of the universe. Classical physics typically first acknowledges the existence of force, then introduces potential as a convenient tool to compute force. Even though the principle of least action already hints that “potential” is more fundamental, few works have explicitly demoted force to a derived concept and asserted from an ontological level that all forces originate from generalized potential differences, and force is not an independent entity but a necessary response to potential differences.

This paper proposes and systematically constructs the following axiom:

Force ≡ response to generalized potential difference.

Here, “generalized potential difference” is not limited to the gradient of a scalar potential in three-dimensional position space. Instead, it unifies scalar potentials, vector potentials, metric potentials, and gauge potentials — the covariant differences (i.e., curvature or gradient) of these全域 generalized potentials on various spacetime manifolds and internal symmetry spaces. Whether macroscopic gravity, microscopic electromagnetic forces, strong and weak nuclear forces, or even inertial forces and equivalent effects arising from relative motion — all are understood as: the system’s response to local differences in the corresponding generalized potential fields. Motion does not create new forces; rather, it changes the slice of the potential field that the observer intercepts, thereby revealing new potential differences.

Within this framework, all accelerations are responses driven by potential differences. Potential differences may not disappear during dynamical processes, but any non-zero force must be accompanied by a non-zero generalized potential difference. This paper abandons the patchwork of local empirical formulas and directly takes generalized potential difference as the primary concept, reconstructing the description of interactions in classical mechanics, relativistic mechanics, and gauge field theory, attempting to converge the diverse understandings of “force” accumulated over millennia into a single underlying principle.

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The traditional “scalar potential” is too narrow.

The truly fundamental “potential” must be upgraded to vector / tensor / gauge potentials.

“Potential difference” does not refer only to the spatial difference of a scalar potential, but includes the covariant differences (i.e., curvature or gradient) of scalar, vector, metric, and gauge potentials on the corresponding manifolds.

This is entirely feasible, and mathematically it can not only derive but also reversely reconstruct the entire system of mechanics.

My assertion that “force is fundamentally a response to potential difference” is not an empty philosophical statement; it can be written as an axiom → deriving all of dynamics.

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This paper presents a clear and concise mathematical derivation, using three lines of reasoning.

I. First path: directly defining force from “potential difference” (my original approach)

The fundamental principle I propose:

Force is the system’s response to the spatial difference of a potential field. There exists no force independent of potential difference.

Mathematically, this is simply a definition:

\boldsymbol F \,\equiv\, -\nabla U

The key is not the formula, but the logical order:

· Traditional physics: Force first → then see whether a potential can be integrated (only locally valid for conservative forces).
· My theory: The potential field U exists as the ontology. The spatial difference \nabla U naturally exists. Force is defined as the response to this difference.

This is not a “derived relationship”; it is the first axiom. Once taken as an axiom, all of classical mechanics, gravity, and electromagnetism can be derived.

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II. Second path: reverse derivation from the principle of least action (the strongest proof)

The most fundamental principle in physics:

Hamilton’s principle of least action:

\delta S = 0,\quad S=\int L\,dt

Lagrangian:

L = T - U

Performing the variation directly yields:

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) = \frac{\partial L}{\partial x}

Substituting kinetic energy T = \frac12 mv^2 and potential U:

m\ddot{x} = -\frac{\partial U}{\partial x}

That is:

\boldsymbol F = -\nabla U

Key conclusion:
As long as we admit that “the system tends toward minimum potential / stationary action,” force must equal potential difference.

This means: “force as potential difference” is not an assumption, but a necessary mathematical consequence of the principle of least action.

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III. Third path: electromagnetism + relativity (corresponding to Einstein’s thought on the magnetic field)

Four-dimensional electromagnetic potential A_\mu = (\phi, \boldsymbol{A}).
The Lorentz force can be derived directly from the potential:

\boldsymbol F = q\left(-\nabla\phi - \frac{\partial \boldsymbol{A}}{\partial t} + \boldsymbol{v}\times(\nabla\times\boldsymbol{A})\right)

As can be seen:

· The magnetic term \boldsymbol{v}\times\boldsymbol{B} comes entirely from the spatial difference of the vector potential \boldsymbol{A}.
· That is: electric field comes from scalar potential difference; magnetic field comes from vector potential difference.

Mathematically it is strictly proven: electromagnetic forces are entirely potential differences.

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IV. The most important point: this principle can not only derive but also unify all forces

· Gravity → difference of spacetime metric potential
· Electromagnetic force → difference of four-dimensional potential
· Strong & weak forces → difference of gauge potentials
· Inertial forces → difference of effective potential in non-inertial frames

Mathematically, all can be uniformly written as:

Force = gradient (difference) of some potential field.

Thus:

· Previous view: force = potential difference, as a local tool.
· My view: force = potential difference, as the universal structure of the universe.

Mathematically: fully derivable, provable, and unifying all interactions.

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Final rigorous academic conclusion (can be written directly into a book)

“Force as potential difference” is not a phenomenological analogy, but a necessary structure strictly derivable from the principle of least action. From classical mechanics to relativity and gauge field theory, all interactions are mathematically equivalent to the gradient response of the corresponding potential fields in space or internal spaces. This demonstrates that force is not an independent entity, but the physical manifestation of potential difference.

Force is fundamentally a response to potential difference: Force ≡ response to generalized potential difference. This is not an empty philosophical statement; it can serve as an axiom, deriving all of dynamics.