[Paper Review] To string together six theorems of physics by Pythagoras theorem

H. Y. Cui|ArXiv.org|May 8, 2002

Cancer and biochemical research4 references3 citations

TL;DR

This paper proposes that Pythagoras' theorem serves as a foundational axiom from which six core theorems in physics—Newton’s three laws, universal gravitation, Coulomb’s law, and relativistic dynamics—can be rigorously derived through geometric and kinematic reasoning. By treating the Pythagorean relation in spacetime as fundamental, the author establishes a unified axiomatic framework that reveals deep internal connections among classical and relativistic mechanics, offering a new pedagogical and conceptual foundation for physics education.

ABSTRACT

In this paper, we point out that there are at lest six theorems in physics sharing common virtue of Pythagoras theorem, so that it is possible to string these theorems together with the Pythagoras theorem for physics teaching, the six theorems are Newton's three laws of motion, universal gravitational force, Coulomb's law, and the formula of relativistic dynamics. Knowing the internal relationships between them, which have never been clearly revealed by other author, will benefit the logic of physics teaching.

Motivation & Objective

• To reveal previously unrecognized internal relationships among six major theorems in physics by deriving them from a single mathematical axiom: Pythagoras' theorem.
• To establish a unified axiomatic framework for mechanics based on the geometric validity of Pythagoras' theorem in inertial frames.
• To reframe physics education by stringing together Newton’s laws, gravity, electromagnetism, and relativistic dynamics through a common geometric foundation.
• To challenge the conventional view of Newtonian mechanics as an approximation by showing it is equally valid as relativistic mechanics when derived from the same geometric principle.

Proposed method

• Assume Pythagoras’ theorem holds in inertial reference frames as the foundational axiom, expressed as $ \Delta l^2 = \Delta x_1^2 + \Delta x_2^2 + \Delta x_3^2 $.
• Derive Newton’s second law $ \mathbf{f} = \frac{d(m\mathbf{v})}{dt} $ and the kinetic energy theorem $ \frac{d}{dt}(\frac{1}{2}mv^2) = \mathbf{f} \cdot \mathbf{v} $ from the time derivative of the Pythagorean velocity relation.
• Extend the derivation to composite systems to derive Newton’s first and third laws via conservation of momentum and force balance.
• Use the vector cross product of position and velocity to derive angular momentum conservation $ \mathbf{r} \times \mathbf{v} = \mathbf{h} = \text{const} $, leading to central force laws.
• Expand the force in a Taylor series in $ 1/r $, and integrate to derive potential energy forms, showing that inverse-square and other laws emerge naturally.
• Generalize the framework to relativistic mechanics by introducing 4-vectors and showing that the relativistic dynamics and force-velocity orthogonality emerge from the same Pythagorean structure.

Experimental results

Research questions

• RQ1 Can Newton’s three laws of motion be derived from the geometric postulate of Pythagoras’ theorem in inertial frames?
• RQ2 How do universal gravitation and Coulomb’s law emerge from the same geometric foundation as Newtonian mechanics?
• RQ3 What is the relationship between relativistic dynamics and Newtonian mechanics when both are derived from Pythagoras’ theorem?
• RQ4 Why does nature appear to favor relativistic mechanics over Newtonian mechanics, despite both being derivable from the same axiom?
• RQ5 Can an axiomatic system of physics be built solely from Pythagoras’ theorem, eliminating the need for multiple independent postulates?

Key findings

• Newton’s second law and the kinetic energy theorem are derived directly from the time derivative of the Pythagorean velocity relation $ v^2 = v_1^2 + v_2^2 + v_3^2 $.
• Newton’s first and third laws are derived from the same geometric axiom when applied to composite systems, via momentum conservation and force balance.
• The universal gravitational force and Coulomb’s law are shown to emerge from the requirement of central forces and angular momentum conservation, derived from the Pythagorean structure.
• The relativistic dynamics formula, including the 4-vector formulation $ \widetilde{v}_\mu \widetilde{f}^\mu = 0 $, is derived from the same geometric principle, showing consistency with the relativistic limit.
• The paper demonstrates that both Newtonian and relativistic mechanics are equally valid when derived from Pythagoras’ theorem, with Newtonian mechanics emerging as a limit when the speed of light becomes infinite.
• The inertial frame of reference is redefined as the frame in which Pythagoras’ theorem holds, offering a new conceptual basis for the foundations of mechanics.