Revisiting Kepler: new symmetries of an old problem

The $Kepler$ $problem$ studies the planar motion of a point mass subject to a central force whose strength varies as the inverse square of the distance to a fixed attracting center. The orbits form a 3-parameter family of unparametrized plane curves, consisting of all conics sharing a focus at the attracting center. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits form a `flat family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves (a result of S. Lie). The new symmetries are different from the well-studied `hidden symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the problem. Furthermore, each 2-parameter family of Kepler orbits with fixed non-zero energy admits $mathrm{PSL}_2(mathbb{R})$ as its symmetry group and coincides with one of the items of a classification due to A. Tresse (1896) of 2nd order ODEs admitting a 3-dimensional group of point symmetries. Other items on Tresse list also appear in Keplers problem by considering repulsive instead of attractive force or motion on a surface with (non-zero) constant curvature. Underlying these newly found symmetries is a duality between Keplers plane and Minkowskis 3-space parametrizing the space of Kepler orbits.