Goursat distribution and sub-Riemannian structures

We obtain the Lie group whose action leaves invariant the sub-Riemannian structures associated with Goursat systems and Euclidean metrics. The group naturally contains the Heisenberg group, the nilpotent group associated with the Martinet case, and the group corresponding to systems of Engel type. We compute also the Casimir functions of the associated nilpotent Poisson algebra. Our results generalize previous works on this problem of nonholonomic systems. A particular physical problem described by our model is the motion of electric charges in certain static inhomogeneous magnetic fields. We define a new algebraic curve in total space and compute two examples of sub-Riemannian extremals in cotangent space. (C) 2003 American Institute of Physics.