Application of codimension one foliation in Zermelo's problem on Riemannian manifolds

摘要

We research the Zermelo navigation problem on Riemannian manifolds in dim(? × M) = 3 under the force representing the action of the perturbing "wind" distribution modeled by the vector field on manifold M. We consider a fibered manifold π: ? × M → ? where M represents the "sea", ? represents the time and π is the first canonical projection. We perturb it by a time-(in)dependent vector field on the manifold M with a codimension one foliation. We aim to research the behavior of the paths in reference to their optimization. Geometrically we find the deviation of geodesics under the action of a time-dependent vector field. We present corresponding simulation with the whirlpool perturbation. We also consider a new concept of real application in the navigational decision support systems.