Foundational Arithmetical Pattern Equations in Kinematics and Dynamics

Abstract

This study explores how arithmetic models can help to clarify the underlying structure of qualitative laws in mathematical physics. Previous research via formal logic and rational analysis uncovered fundamental methodological constraints within continuous deterministic classical mechanics. It's determinism in the absence of a discrete model, related to Soley's continuous calculus, that has obscured both the theoretical and mathematical basis. Classical mechanics does not possess a mathematically distinct representative model; instead, it is qualitatively divided into two core principles: constant motion (the law of inertia) and accelerated motion caused by an unbalanced force (the second law of motion). Conversely, these motions inherently require an autonomous model. Compatibly, the arithmetic pattern is carefully chosen due to its discrete and continuous frameworks. In this context, variables under constant motion follow a deterministic continuous model, while variables of accelerated motion adhere to a deterministic discrete model. Furthermore, this dual framework provides a novel mathematical foundation for the law of inertia; and, although qualitatively constant and accelerated motion are identified with their autonomous laws, mathematically, they can be unified through total energy conservation in both uniform and variable acceleration motion in the presence of friction. This study enhances conceptual clarity, simplifies mathematical representation, and improves the accuracy of models in kinematics and dynamics, yielding a more intuitive and rigorous framework that better aligns with observed physical law.

Key words: Deterministic, Discrete, Continuous, Calculus, Mechanics, Arithmetic.