Séminaire de Systèmes Dynamiques

The Kepler problem - new symmetries of an old system

by Gil Bor (CIMAT, Guanajuato, Mexico)

207 (Bat 1R2)


Bat 1R2

The Kepler problem, introduced by Newton in 1687, is a system of second order differential equations describing the motion of a planet around the sun. The orbits of the motion form a 3-parameter family of plane curves, consisting of all conics  (ellipses, parabolas and hyperbolas) sharing a focus at a fixed point (the sun). A rich structure emerges, with a 7-dimensional group of symmetries for the full family, the maximum dimension possible for a 3-parameter family of plane curves, a 3-dimensional group for the 2-parameter family of orbit of fixed  energy, and an 8-dimensional group for Kepler parabolas or orbits of fixed non-zero angular momentum. Underlying these symmetries is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits.

Joint work with Connor Jackman (Heidelberg). Reference: arxiv.org/abs/2106.02823