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Application of codimension one foliation in Zermelo’s problem on Riemannian manifolds. (English) Zbl 1334.53010

Summary: We research the Zermelo navigation problem on Riemannian manifolds in \(\dim(\mathbb{R}\times M)=3\) under the force representing the action of the perturbing “wind” distribution modeled by the vector field on manifold \(M\). We consider a fibered manifold \(\pi:\mathbb{R}\times M\to\mathbb{R}\) representing the “sea”, \(\mathbb{R}\) represents the time and \(\pi\) is the first canonical projection. We perturb it by a time-(in)dependent vector field on the manifold \(M\) with a codimension one foliation. We aim to research the behavior of the paths in reference to their optimization. Geometrically we find the deviation of geodesics under the action of a time-dependent vector field. We present corresponding simulation with the whirlpool perturbation. We also consider a new concept of real application in the navigational decision support systems.

MSC:

53B20 Local Riemannian geometry
53A55 Differential invariants (local theory), geometric objects
53C12 Foliations (differential geometric aspects)
49J53 Set-valued and variational analysis
57R30 Foliations in differential topology; geometric theory
53B50 Applications of local differential geometry to the sciences
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[1] Bao, D.; Robles, C.; Shen, Z., Zermelo navigation on Riemannian manifolds, J. Differ. Geom., 66, 377-435 (2004) · Zbl 1078.53073
[2] Bijlsma, S. J., Minimal time route computation for ships with pre-specified voyage fuel consumption, J. Navig., 61, 723-733 (2008)
[3] Bijlsma, S. J., Optimal aircraft routing in general wind fields, J. Guid. Control Dyn., 32, 1025-1029 (2009)
[4] Candel, A.; Conlon, L., Foliations. I, Grad. Stud. Math., vol. 23 (2000), American Mathematical Society · Zbl 0936.57001
[5] Candel, A.; Conlon, L., Foliations. II, Grad. Stud. Math., vol. 60 (2003), American Mathematical Society · Zbl 1035.57001
[6] Caratheodory, C., Calculus of Variations and Partial Differential Equations of the First Order (1935), American Mathematical Society, Chelsea Publishing, reprinted in 2008
[7] Chern, S. S.; Shen, Z., Riemann-Finsler Geometry, Nankai Tracts in Mathematics (2005), World Scientific: World Scientific River Edge (N.J.), London, Singapore · Zbl 1085.53066
[8] Jardin, M.; Bryson, A., Methods for computing minimum - time paths in strong winds, J. Guid. Control Dyn., 35, 165-171 (2012)
[9] Kopacz, P., Proposal on a Riemannian approach to path modeling in a navigational decision support system, (Mikulski, J., Activities of Transport Telematics. Activities of Transport Telematics, Commun. Comput. Inf. Sci., vol. 395 (2013), Springer: Springer Berlin, Heidelberg), 294-302
[10] Kopacz, P., Simulation of Zermelo navigation on Riemannian manifolds for \(\dim(R \times M) = 3\), (Navigational Problems: Marine Navigation and Safety of Sea Transportation (2013), A. Balkema Book, CRC Press, Taylor & Francis Group: A. Balkema Book, CRC Press, Taylor & Francis Group Boca Raton - London - New York - Leiden), 333-337
[11] Li, B.; Xu, C.; Teo, K. L.; Chu, J., Time optimal Zermelo’s navigation problem with moving and fixed obstacles, Appl. Math. Comput., 224, 866-875 (2013) · Zbl 1337.49071
[12] Molino, P., Riemannian Foliations, Prog. Math., vol. 73 (1988), Birkhäuser
[13] Palacek, R.; Krupkova, O., On the Zermelo problem in Riemannian manifolds, Balk. J. Geom. Appl., 17, 77-81 (2012) · Zbl 1284.53015
[14] Rovenski, V.; Wolak, R., Deforming metrics of foliations, Cent. Eur. J. Math., 11, 1039-1055 (2013) · Zbl 1275.53062
[15] Tondeur, P., Geometry of Foliations, Monographs in Mathematics, vol. 90 (1997), Birkhäuser Verlag · Zbl 0905.53002
[16] Wolak, R., Foliated and associated geometric structures on foliated manifolds, Ann. Fac. Sci. Toulouse, 10, 337-360 (1989) · Zbl 0698.57007
[17] Zermelo, E., Über das Navigationsproblem bei ruhender oder veranderlicher Windverteilung, Z. Angew. Math. Mech., 11, 114-124 (1931) · Zbl 0001.34101
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