Exploration of Intrinsic Kinematics

Coordinate-Free Acceleration Direction in Two and Three Dimensions

Abstract

This work presents a coordinate-free (intrinsic) derivation of the standard tangential–normal decomposition of acceleration for a particle moving along a smooth planar curve. Our decomposition of acceleration components is classical; the main emphasis is the conceptual distinction between the local acceleration tilt ϕ (intrinsic, measurable without coordinates) and the global path orientation Θ(s) (extrinsic, relative to a fixed reference ex. x/y axis). Using the Frenet-Serret relations and curvature κ, we derive an exact expression for ϕ and show that Θ satisfies the differential relation dΘ/ds = κ(s). After integrating this relation, we obtain an intrinsic–extrinsic formula for the particle’s absolute acceleration direction entirely in terms of curvature and kinematic rates (κ, s˙, s¨). We then apply the formula to an example problem, and verify the results through standard Cartesian methods to confirm consistency in concept. This intrinsic viewpoint offers a new, and simpler, outlook on precise geometric distinctions through a "void-analogy" to cover overlooked geometric distinctions in classical treatments of planar motion.