This messadge contains a discription of the new family of adaptive symplectic conservative numerical methods for the Kepler problem. The methods perform symplectic mapping from the initial to the current state and, therefore, they preserve phase volume. In contrast to the existing symplectic methods, e.g., Verlet integrator, they preserve all first integrals of the Kepler problem i.e., angular momentum, full energy and Laplace-Runge-Lenz vector in the frame of the exact arithmetic. The orbit and the velocity hodograph are preserved as well. The numerical integration adaptive step is chosen automatically based on the local features of the solution. The step decreases where phase variables change most rapidly. The methods approximate dependence of phase variables on time with 2-nd or 4-th, or 6-th order.

Hamiltonian system, Kepler problem, symplectic integrators, adaptive methods, first integrals, solution parametrization, order of accuracy

Elenin Georgii Georgievich,  Elenin2@rambler.ru,  orcid.org/0000-0002-1708-9406,  Lomonosov Moscow State University