On conservative discretization of the Kepler problem

Abstract

In this note we consider conservative discretizations of the Kepler motion. They are based on the L-transformations (a generalization of Levi-Civita and Kustaanheimo–Stiefel transformations, which are used for the two-and three-dimensional Kepler motion respectively). The discretizations preserve all first integrals of the Kepler motion: the Hamiltonian function, the angular momentum and Runge-Lenz vector. Conservation of the first integrals is proved with the help of L-matrix identities. There exist higher order schemes as well as the exact integrator of the Kepler motion. The proposed schemes permit explicit implementation.