Projective dynamics and classical gravitation

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.

20 pages, 4 figures

References in corpus (2)

• Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$
• Generalized problem of two and four Newtonian centers

Cited by in corpus (10)

• Examples of integrable and non-integrable systems on singular symplectic manifolds
• Projective dynamics and first integrals
• Keplerian Dynamics on the Heisenberg Group and Elsewhere
• Periodic Orbits in the Kepler-Heisenberg Problem
• Orbits in the problem of two fixed centers on the sphere
• Darboux inversions of the Kepler problem
• Lambert's theorem and projective dynamics
• Projective Dynamics and an Integrable Boltzmann Billiard Model
• Revisiting Kepler: new symmetries of an old problem
• Conelike radiant structures