Projective dynamics and classical gravitation

We show that there exists a projective dynamics of a particle. It underlies intrinsically the classical particle dynamics as projective geometry underlies Euclidean geometry. In classical particle dynamics a particle moves in the Euclidean space subjected to a potential. In projective dynamics the position space has only the local structure of the real projective space. The particle is subjected to a field of projective forces. A projective force is not an element of the tangent bundle to the position space, but of some fibre bundle isomorphic to the tangent bundle. These statements are direct consequences of Appell’s remarks on the homography in mechanics, and are compatible with similar statements due to Tabachnikov concerning projective billiards. When we study Euclidean geometry we meet some particular properties that we recognize as projective properties. The same is true for the dynamics of a particle. We show that two properties in classical particle dynamics are projective properties. The fact that the Keplerian orbits close after one turn is a consequence of a more general projective statement. The fact that the fields of gravitational forces are divergence free is a projective property of these fields.