Symplectic integrators in sub-Riemannian geometry: the Martinet case
  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. 
Monique Chyba, Ernst Hairer, Gilles Vilmart (IRISA / INRIA Rennes)