On generalization of Zermelo navigation problem on Riemannian manifolds

TL;DR

This paper extends the Zermelo navigation problem to Riemannian manifolds with variable ship speed and perturbations, utilizing Randers-type Finsler metrics to find solutions.

Contribution

It generalizes the classical Zermelo navigation problem to more complex Riemannian settings with variable speeds and perturbations, providing new mathematical insights.

Findings

Formulation of the navigation problem on Riemannian manifolds with variable speed.

Application of Randers-type Finsler metrics to solve the generalized problem.

Potential framework for navigation in complex geometric environments.

Abstract

We generalize the Zermelo navigation problem and its solution on Riemannian manifolds $(M, h)$ admitting a space dependence of a ship's own speed $|u(x)|_h\leq1$ in the presence of a perturbation $W$ determined by a mild velocity vector field $|W(x)|_h<|u(x)|_h$, with application of Finsler metric of Randers type.