The Kepler problem: New Symmetries of an Old Problem

The Kepler orbits form a 3-parameter family of plane curves, consisting of all conics (ellipses, parabolas and hyperbolas) sharing a focus at some fixed point. I will describe the symmetry properties of this family, as well as certain natural 2-parameter subfamilies, such as those of fixed energy or angular momentum.
The standard technique for studying such symmetries is via a PDE system for the infinitesimal symmetries of a 3rd order ODE encoding the 3-parameter family of curves – a well known method due to S. Lie, nowadays usually computer assisted. I will describe instead a natural projective geometric construction, exploiting the geometry of the Minkowski's 3-space parametrizing the space of Kepler orbits.
This is joint work with Connor Jackman (ITAM, Mexico). See arxiv.org/abs/2106.02823