Symplectic integrators in sub-Riemannian geometry: the Martinet case
M. Chyba, E. Hairer, and G. Vilmart
Abstract. 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.
Key Words. Sub-Riemannian geometry, Martinet, abnormal geodesic, symplectic
integrator, backward error analysis.