Zermelo Navigation Problem in Geometry

Abstract

One of the long standing problems in navigation is explained and the mathematical formulation using Riemannian and Finsler geometry is introduced. Randers norms and Randers metrics used for the description of the influence of the wind or of the current on the cruising boat are explained. The explicit formulas relating the vector field representing the wind or the current with the corresponding Randers metric are given. Special situations with complete mathematical solution of the problem are mentioned, namely the case of the Riemannian manifold with constant sectional curvature, Finsler spaces with constant flag curvature, some special manifolds in dimension 3 and other. Some possibilities for different mathematical approach to the problem and also for further research are indicated.

Keywords

• Riemannian manifold
• Finsler space
• Geodesic
• Zermelo navigation problem