  We compare the performances of symplectic and non-symplectic integrators for
the computation of normal geodesics and conjugate points in sub-Riemannian
geometry at the example of the Martinet case. For this case study we consider
first the flat metric, and then a one parameter perturbation leading to non
integrable geodesics. From our computations we deduce that near the abnormal
directions a symplectic method is much more efficient for this optimal control
problem. The explanation relies on the theory of backward error analysis in
geometric numerical integration.
