Projective dynamics and classical gravitation

TL;DR

This paper develops a projective dynamics framework on a vector space, revealing new symmetries in Kepler's problem and showing that divergence-free force fields exist only in four dimensions.

Contribution

It introduces a novel formulation of dynamics using projective geometry, providing new insights into Kepler's problem and the nature of force fields without relying on metrics.

Findings

Trajectories are not parameterized in projective dynamics.

The Kepler problem corresponds to a simple force field with conic sections.

Divergence-free projective force fields exist only in four-dimensional spaces.

Abstract

Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics such that the position of the particle is a "ray" i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is a 2-plane containing the ray. Throwing the particle with a given initial impulsion defines a projective trajectory. It is a curve in the space of rays S(V), together with an impulsion attached to each ray. In the simplest example where the force is identically zero, the curve is a straight line and the impulsion a constant bivector. A striking feature of projective dynamics appears: the trajectories are not parameterized. Among the projective force fields corresponding to a central force, the one defining the Kepler problem is simpler than those corresponding to other homogeneities. Here the thrown ray describes a quadratic cone whose section by a hyperplane corresponds to a Keplerian conic. An original point of view on the hidden symmetries of the Kepler problem emerges, and clarifies some remarks due to Halphen and Appell. We also get the unexpected conclusion that there exists a notion of divergence-free field of projective forces if and only if dim V=4. No metric is involved in the axioms of projective dynamics.