  Phase fitting has been extensively used during the last years to improve the
behaviour of numerical integrators on oscillatory problems. In this work, the
benefits of the phase fitting technique are embedded in discrete Lagrangian
integrators. The results show improved accuracy and total energy behaviour in
Hamiltonian systems. Numerical tests on the long term integration (100000
periods) of the 2-body problem with eccentricity even up to 0.95 show the
efficiency of the proposed approach. Finally, based on a geometrical evaluation
of the frequency of the problem, a new technique for adaptive error control is
presented.
