The objective of this article is to present geometric and numerical techniques developed
to study the orbit transfer between Keplerian elliptic orbits in the two-body problem or between
quasi-Keplerian orbits in the Earth-Moon transfer when low propulsion is used. We concentrate our
study on the energy minimization problem. From Pontryagin’s maximum principle, the optimal solution
can be found solving the shooting equation for smooth Hamiltonian dynamics. A first step in
the analysis is to find in the Kepler case an analytical solution for the averaged Hamiltonian, which
corresponds to a Riemannian metric. This will allow to compute the solution for the original Kepler
problem, using a numerical continuation method where the smoothness of the path is related to the
conjugate point condition. Similarly, the solution of the Earth-Moon transfer is computed using
geometric and numerical continuation techniques.