A conservative discretization of the Kepler problem based on the L-transformations

Conservative numerical schemes for the two- and three-dimensional Kepler motion were recently proposed in [Y. Minesaki, Y. Nakamura, Phys. Lett. A 306 (2–3) (2002) 127; Y. Minesaki, Y. Nakamura, Phys. Lett. A 324 (4) (2004) 282]. They are based on Levi-Civita and Kustaanheimo–Stiefel transformations respectively. The schemes preserve all first integrals of Kepler motion: the Hamiltonian function, the angular momentum and Runge–Lenz vector. In the present Letter we extend this approach to the L-transformations (a generalization of Levi-Civita and Kustaanheimo–Stiefel transformations). Thus, we can consider more general family of the conservative numerical schemes for the two- and three-dimensional Kepler motion as well as construct conservative schemes for higher-dimensional Kepler problems. Conservation of the first integrals is proved with the help of L-matrix identities. The proposed numerical scheme permits explicit implementation.