Abstract: Consider small bodies orbiting around the Sun on Keplerian
orbits, under a Newtonian attraction. Consider this known property: Two
such orbits may have at most two intersections. Note that another
attraction law may allow infinitely many intersections between two
orbits. We will recall a short proof of the property which confirms the
effectiveness of an equation advertised by Gauss and used, for example,
in the article by Bor and Jackman in 2023.
We will present a more general attraction law due to Jacobi where this
property and many others survive. Its definition does not use a
Euclidean structure of the space. So, for example, there is no
invariance by rotation, while there is an "angular momentum" which is
conserved. There is also an "eccentricity vector". We will explain a
simple existence result for closed orbits obtained with Antonio Ureña,
which is related with a convexity property.