Riemannian Metric of the Averaged Controlled Kepler Equation

A non-autonomous sub-Riemannian problem is considered: Since periodicity with respect to the independent variable is assumed, one can define the averaged problem. In the case of the minimization of the energy, the averaged Hamiltonian remains quadratic in the adjoint variable. When it is non-degenerate, a Riemannian problem and the corresponding metric can be uniquely associated to the averaged problem modulo the orthogonal group of the quadratic form. The analysis is applied to the controlled Kepler equation. Explicit computations provide the averaged Hamiltonian of the Kepler motion in the threedimensional case. The Riemannian metric is given, and the curvature of a special subsytem is evaluated. keywords: periodic sub-Riemannian problems, averaging, Riemannian metrics, minimum energy control, Kepler equation