How Much Geometry Does Newtonian Dynamics Fix?

Abstract

Newtonian mechanics is usually written on a spacetime of fixed geometry. Here we build that geometry, introducing a propagator and a bilinear form only as the description demands. Within this reconstruction the Principle of Inertia emerges as a theorem rather than an independent postulate. Inertial motion is blind to the scale of the form and to the torsion the propagator defines, and that blindness is a geometric precursor of the \emph{universality of free fall}, before gravity enters. Newton's Second Law is a compatibility between two derivatives along a curve. The force exceeds the mass times the lowered acceleration by an exact term, and the law survives arbitrary torsion. Requiring force-free motion to be inertial along curves parametrised by arc length restricts the torsion to be totally antisymmetric. The remaining torsion is a three-form which we postulate the particle carries along its worldline. Its contraction with the velocity annihilates that velocity, so the supplementary condition of spinning-particle mechanics holds identically, and we propose that contraction as a classical spin. The algebra is familiar from relativistic spin hydrodynamics; the assignment, to the best of our knowledge, has not been proposed elsewhere. Since inertial motion resolves neither the mass nor the spin, the predictive content lies in the transport law we postulate, which turns the spin against a parallel-transported frame at a rate fixed by the acceleration alone and by no coupling constant. That rate becomes a laboratory number only once the arc-length parameter is identified with a clock, which we leave open. The dynamics asks non-degeneracy of the form, so this spacetime is not Newton--Cartan's Galilean one, and gravitation lies outside the construction. We prove every mathematical result, make three constitutive postulates and one physical conjecture.