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Piotr Kopacz

Publications and source records attributed to Piotr Kopacz.

8 recordsLinked to original sources
arXiv2026

Superwind and navigation of least time on Riemannian manifolds

In this work, minimum-time navigation on Riemannian manifolds is analyzed by means of Finsler geometry. The introduced notion of a superwind being anisotropic, reducible, and unknown a priori encompasses a wind in the spirit of Zermelo's navigation, and a gravitational wind in the sense of a slippery slope model, including Matsumoto's navigation on a mountain slope as the particular cases. The considered generalization also deploys a non-uniform slope, where the counterbalance to the transverse impact of superwind causing a lateral drift varies over space. This requires in particular an extension of the existing theory of general $(α, β)$-metrics, which is presented in our study. The solution of the navigation problem is given by a new Finsler metric and the time-minimizing geodesics, which are determined. Moreover, the thorough analysis establishes the necessary and sufficient conditions for strong convexity under which the resultant velocity defines a Finsler metric. For completeness, a variety of comprehensive and comparative examples are explored in dimension two.

math.DG↗
arXiv2025

A general model for time-minimizing navigation on a mountain slope under gravity

In this work, we solve the generalized Matsumoto's slope-of-a-mountain problem by means of Riemann-Finsler geometry, making close links with the Zermelo navigation problem. The time-minimizing navigation under gravity is analyzed in the general model of a slippery mountain slope being a Riemannian manifold. Both the transverse and longitudinal gravity-additives with respect to the direction of motion are admitted to vary simultaneously in the full ranges, showing the impact of cross- and along-traction on the slippery slope. By the anisotropic deformation of the background Riemannian metric and rigid translation with the use of the rescaled gravitational wind, we obtain the purely geometric solution for optimal navigation, which is given by a new Finsler metric belonging to the class of general $(α, β)$-metrics. The related strong convexity conditions are established and time geodesics are described. Moreover, the evolution of time fronts and the behavior of time-minimizing trajectories in relation to various gravity effects on the slippery slope, gravitational wind force and direction of motion are thoroughly discussed and visualized by several two-dimensional examples.

math.DG↗
arXiv2024

Time geodesics on a slippery cross slope under gravitational wind

In this work, we pose and solve the time-optimal navigation problem considered on a slippery mountain slope modeled by a Riemannian manifold of an arbitrary dimension, under the action of a cross gravitational wind. The impact of both lateral and longitudinal components of gravitational wind on the time geodesics is discussed. The varying along-gravity effect depends on traction in the presented model, whereas the cross-gravity additive is taken entirely in the equations of motion, for any direction and gravity force. We obtain the conditions for strong convexity and the purely geometric solution to the problem is given by a new Finsler metric, which belongs to the type of general $(α, β)$-metrics. The proposed model enables us to create a direct link between the Zermelo navigation problem and the slope-of-a-mountain problem under the action of a cross gravitational wind. Moreover, the behavior of the Finslerian indicatrices and time-minimizing trajectories in relation to the traction coefficient and gravitational wind force are explained and illustrated by a few examples in dimension two. This also compares the corresponding solutions on the slippery slopes under various cross- and along-gravity effects, including the classical Matsumoto's slope-of-a-mountain problem and Zermelo's navigation.

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arXiv2016

Generalized Zermelo navigation on Hermitian manifolds under mild wind

We generalize and study the Zermelo navigation problem on Hermitian manifolds in the presence of a perturbation $W$ determined by a mild complex velocity vector field $||W(z)||_h<||u(z)||_h$, with application of complex Finsler metric of complex Randers type. By admitting space-dependence of ship's relative speed $||u(z)||_h\leq1$ we discuss the projectively related complex Finsler metrics, the geodesics corresponding to the solutions of Zermelo's problem, in particular the conformal case and the connections between the corresponding background Hermitian metric $h$, new Hermitian metric $a$ and resulting complex Randers metric $F$. Moreover, we present some necessary and sufficient conditions for the obtained locally projectively flat solutions. Our findings are also illustrated with several examples.

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arXiv2016

On generalization of Zermelo navigation problem on Riemannian manifolds

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.

math.DG↗
arXiv2016

A note on time-optimal paths on perturbed spheroid

We consider the Zermelo navigation problem on the ellipsoid of revolution (spheroid) in the presence of a perturbation $W$ determined by a mild velocity vector field, $|W|<1$, with application of Finsler metric of Randers type in the context of the corresponding optimal control represented by a time-efficient ship's heading $φ(t)$ (steering direction). As the example we present the solutions to the problem on an oblate ellipsoid with acting infinitesimal rotation.

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