The Geometry of Motions, Vol. I: Mechanics as Geometries

TL;DR

This paper offers a group-theoretic geometric interpretation of the evolution of mechanics, linking each fundamental theory to a specific geometry defined by its invariance group, and clarifies the nature of key epistemological shifts in physics.

Contribution

It introduces a unified geometric framework that characterizes the historical transitions in mechanics through the lens of group theory and invariance groups.

Findings

Identifies the geometries associated with Aristotelian, Galilean, and Poincaré groups.

Proves the non-existence of Galilean-invariant space and Poincaré-invariant time.

Provides a rigorous mathematical distinction between primary and secondary epistemological ruptures.

Abstract

This work presents a group-theoretic interpretation of the historical evolution of mechanics, proposing that each fundamental theory of motion corresponds to a distinct geometry in the sense of Felix Klein. The character of each geometry is uniquely determined by its Inertia Group-the group of spacetime transformations that preserves its privileged class of inertial motions. We trace the three major epistemological ruptures in the history of mechanics by translating foundational physical principles into the unambiguous language of group theory. The analysis begins with Aristotelian mechanics, where absolute Space and Time are shown to be homogeneous spaces of the Group of Aristotle, defined by its preservation of rest. The second rupture, driven by the relativity of motion (Bruno, Galileo), leads to the abandonment of absolute Space and the construction of the Galilean Group, which preserves the class of uniform rectilinear motions. The final rupture, precipitated by the crisis in electromagnetism, results in the dissolution of absolute Time and the emergence of the Poincar\'e Group, preserving affine lines and the Minkowski metric. Central results of this approach are theorems demonstrating, with mathematical certainty, the non-existence of a Galilean-invariant "Space" and a Poincar\'e-invariant "Time," where invariance is defined with respect to the action of the corresponding inertia group. This geometric framework provides a unified perspective on the transition from classical to modern physics and allows for a rigorous distinction between 'primary' epistemological ruptures, which alter the Inertia Group itself, and 'secondary' ruptures, which introduce new formalisms for the management of dynamics within a stable geometry.