Time-dependent Zermelo navigation with tacking
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909731066478592 |
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| author | Markvorsen, Steen Pendás-Recondo, Enrique Rygaard, Frederik Möbius |
| author_facet | Markvorsen, Steen Pendás-Recondo, Enrique Rygaard, Frederik Möbius |
| contents | We address the time- and position-dependent Zermelo navigation problem within the framework of Lorentz-Finsler geometry. Since the initial work of E. Zermelo, the task is to find the time-minimizing trajectory between two regions for a moving object whose speed profile depends on time, position and direction. We give a step-by-step review of the classical formulation of the problem, where the geometric shape generated by the velocity vectors -- the speed profile indicatrix -- is strongly convex at each point. We derive new global results for the cases where the indicatrix field is only time-dependent. In such (meso-scale realistic) cases, Zermelo navigation exhibits particularly favorable properties that have not been previously explored, making them especially appealing for both theoretical and numerical investigations. Moreover, motivated by real-world phenomena and examples, we obtain novel results for non-convex (or multi-convex) navigation, i.e. when the indicatrices fail to be convex. In this new setting -- which is not unlike the corresponding Finsler setting for Snell's law -- optimal paths may involve so-called tacking, which stems from discontinuous shifts of direction of motion. The tacking behaviour thus results in zig-zag trajectories, as observed in time-optimal sailboat navigation and surprisingly also in the flight paths of far-ranging seabirds. Finally, we provide new efficient computational algorithms and illustrate the use of them to solve the Zermelo navigation (boundary value) problem, i.e. to find numerically the time-minimizing trajectory between two fixed points in the general non-convex setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_07274 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time-dependent Zermelo navigation with tacking Markvorsen, Steen Pendás-Recondo, Enrique Rygaard, Frederik Möbius Differential Geometry Mathematical Physics 53C22, 53C60, 53B40, 53C80, 65L10 We address the time- and position-dependent Zermelo navigation problem within the framework of Lorentz-Finsler geometry. Since the initial work of E. Zermelo, the task is to find the time-minimizing trajectory between two regions for a moving object whose speed profile depends on time, position and direction. We give a step-by-step review of the classical formulation of the problem, where the geometric shape generated by the velocity vectors -- the speed profile indicatrix -- is strongly convex at each point. We derive new global results for the cases where the indicatrix field is only time-dependent. In such (meso-scale realistic) cases, Zermelo navigation exhibits particularly favorable properties that have not been previously explored, making them especially appealing for both theoretical and numerical investigations. Moreover, motivated by real-world phenomena and examples, we obtain novel results for non-convex (or multi-convex) navigation, i.e. when the indicatrices fail to be convex. In this new setting -- which is not unlike the corresponding Finsler setting for Snell's law -- optimal paths may involve so-called tacking, which stems from discontinuous shifts of direction of motion. The tacking behaviour thus results in zig-zag trajectories, as observed in time-optimal sailboat navigation and surprisingly also in the flight paths of far-ranging seabirds. Finally, we provide new efficient computational algorithms and illustrate the use of them to solve the Zermelo navigation (boundary value) problem, i.e. to find numerically the time-minimizing trajectory between two fixed points in the general non-convex setting. |
| title | Time-dependent Zermelo navigation with tacking |
| topic | Differential Geometry Mathematical Physics 53C22, 53C60, 53B40, 53C80, 65L10 |
| url | https://arxiv.org/abs/2508.07274 |