A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation

TL;DR

This paper extends the Zermelo navigation problem to Riemannian manifolds with variable ship speed and strong perturbations, using Kropina type Finsler metrics to find solutions.

Contribution

It introduces a generalized framework for the Zermelo problem on Riemannian manifolds with strong perturbations and variable speeds, employing Kropina metrics for solutions.

Findings

Generalization of Zermelo navigation problem to manifolds with space-dependent speeds.

Application of Kropina type Finsler metrics to solve the problem.

Potential new insights into navigation under strong perturbations.

Abstract

We generalize the Zermelo navigation problem and its solution on Riemannian manifolds $(M, h)$ admitting a space dependence of a ship's speed $0<|u(x)|_h\leq1$ in the presence of a perturbation $\tilde{W}$ determined by a strong velocity vector field satisfying $|\tilde{W}(x)|_h=|u(x)|_h$, with application of Finsler metric of Kropina type.