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A note on time-optimal paths on perturbed spheroid
1. | Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Prof. St. Lojasiewicza 6, 30 - 348 Kraków, Poland |
2. | Gdynia Maritime University, Faculty of Navigation, Al. Jana Pawla Ⅱ 3, 81-345 Gdynia, Poland |
We consider Zermelo's problem of navigation on a spheroid in the presence of space-dependent perturbation $W$ determined by a weak velocity vector field, $|W|_h<1$. The approach is purely geometric with application of Finsler metric of Randers type making use of the corresponding optimal control represented by a time-minimal ship's heading $\varphi(t)$ (a steering direction). A detailed exposition including investigation of the navigational quantities is provided under a rotational vector field. This demonstrates, in particular, a preservation of the optimal control $\varphi(t)$ of the time-efficient trajectories in the presence and absence of acting perturbation. Such navigational treatment of the problem leads to some simple relations between the background Riemannian and the resulting Finsler geodesics, thought of the deformed Riemannian paths. Also, we show some connections with Clairaut's relation and a collision problem. The study is illustrated with an example considered on an oblate ellipsoid.
References:
[1] |
The International Maritime Organization (IMO), COLREG: Convention on the International Regulations for Preventing Collisions at Sea, London, United Kingdom, 2004. Google Scholar |
[2] |
N. Aldea and P. Kopacz,
Generalized Zermelo navigation on Hermitian manifolds under mild wind, Diff. Geom. Appl., 54 (2017), 325-343.
doi: 10.1016/j.difgeo.2017.05.007. |
[3] |
N. Aldea and P. Kopacz,
Generalized Zermelo navigation on Hermitian manifolds with a critical wind, Results Math., 72 (2017), 2165-2180.
doi: 10.1007/s00025-017-0757-6. |
[4] |
K. J. Arrow, On the use of winds in flight planning, J. Meteor., 6 (1949), 150-159. Google Scholar |
[5] |
D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry (eds. D. Bao et al.), Math. Sci. Res. Inst. Publ., 50 (2004), Cambridge Univ. Press, 197-259. |
[6] |
D. Bao, C. Robles and Z. Shen,
Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435.
|
[7] |
D. C. Brody, G. W. Gibbons and D. M. Meier, Time-optimal navigation through quantum wind,
New J. Phys., 17 (2015), 033048, 8pp.
doi: 10.1088/1367-2630/17/3/033048. |
[8] |
D. C. Brody, G. W. Gibbons and D. M. Meier,
A Riemannian approach to Randers geodesics, J. Geom. Phys., 106 (2016), 98-101.
doi: 10.1016/j.geomphys.2016.03.019. |
[9] |
D. C. Brody and D. M. Meier, Solution to the quantum Zermelo navigation problem, Phys. Rev. Lett., 114 (2015), 100502. Google Scholar |
[10] |
E. Caponio, M. A. Javaloyes and M. Sànchez, Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes, preprint, arXiv: 1407.5494. Google Scholar |
[11] |
C. Carathéodory,
Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965. |
[12] |
S.-S. Chern and Z. Shen,
Riemann-Finsler Geometry, Nankai tracts in mathematics, World Scientific, River Edge (N. J.), London, Singapore, 2005.
doi: 10.1142/5263. |
[13] |
M. A. Earle, Sphere to spheroid comparisons, J. Navigation, 59 (2006), 491-496. Google Scholar |
[14] |
A. de Mira Fernandes, Sul problema brachistocrono di Zermelo, Rendiconti della R. Acc. dei Lincei, XV (1932), 47-52. Google Scholar |
[15] |
C. A. R. Herdeiro, Mira Fernandes and a generalised Zermelo problem: Purely geometric
formulations, Bol. Soc. Port. Mat., 2010, Special Issue. Publication date estimated, 179-191. |
[16] |
M. R. Jardin, Toward Real-Time en Route Air Traffic Control Optimization, Ph. D thesis, Stanford University, 2003. Google Scholar |
[17] |
A. B. Katok,
Ergodic perturbations of degenerate integrable Hamiltonian systems, Math. USSR Izv., 37 (1973), 539-576.
|
[18] |
P. Kopacz, On generalization of Zermelo navigation problem on Riemannian manifolds, preprint, arXiv: 1604.06487. Google Scholar |
[19] |
P. Kopacz,
Application of planar Randers geodesics with river-type perturbation in search models, Appl. Math. Model., 49 (2017), 531-553.
doi: 10.1016/j.apm.2017.05.007. |
[20] |
P. Kopacz,
A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation, An. Sti. U. Ovid. Co.-Mat., 25 (2017), 107-123.
|
[21] |
P. Kopacz,
Application of codimension one foliation in Zermelo's problem on Riemannian manifolds, Diff. Geom. Appl., 35 (2014), 334-349.
doi: 10.1016/j.difgeo.2014.04.007. |
[22] |
P. Kopacz, Proposal on a Riemannian approach to path modeling in a navigational decision support system, in Activities of Transport Telematics. TST 2013. Communications in Computer and Information Science 395 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2013), 294-302. Google Scholar |
[23] |
T. Levi-Civita, Über Zermelo's Luftfahrtproblem, ZAMM-Z. Angew. Math. Me., 11 (1931), 314-322. Google Scholar |
[24] |
B. Manià,
Sopra un problema di navigazione di Zermelo, Math. Ann., 133 (1937), 584-599.
doi: 10.1007/BF01571651. |
[25] |
A. Pallikaris and G. Latsas, New Algorithm for Great Elliptic Sailing (GES), J. Navigation, 62 (2012), 493-507. Google Scholar |
[26] |
A. Pressley,
Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, London, 2010.
doi: 10.1007/978-1-84882-891-9. |
[27] |
C. Robles,
Geodesics in Randers spaces of constant curvature, T. Am. Math. Soc., 359 (2007), 1633-1651.
doi: 10.1090/S0002-9947-06-04051-7. |
[28] |
B. Russell and S. Stepney, Zermelo navigation in the quantum brachistochrone,
J. Phys. A - Math. Theor., 48 (2015), 115303, 29pp.
doi: 10.1088/1751-8113/48/11/115303. |
[29] |
Z. Shen,
Finsler Metrics with ${\bf{K}} = 0$ and ${\bf{S}} = 0$, Can. J. Math., 55 (2003), 112-132.
doi: 10.4153/CJM-2003-005-6. |
[30] |
W.-K. Tseng, M. A. Earle and J.-L. Guo, Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing, J. Navigation, 65 (2012), 549-559. Google Scholar |
[31] |
A. Weintrit and P. Kopacz, Computational algorithms implemented in marine navigation electronic systems, in Telematics in the Transport Environment. TST 2012. Communications in Computer and Information Science 329 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2012), 148-158. Google Scholar |
[32] |
A. Weintrit and P. Kopacz, A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General, in Methods and Algorithms in Navigation (eds. A. Weintrit and T. Neumann), CRC Press, (2011), 123-132. Google Scholar |
[33] |
R. Yoshikawa and S. V. Sabau,
Kropina metrics and Zermelo navigation on Riemannian manifolds, Geom. Dedicata, 171 (2014), 119-148.
doi: 10.1007/s10711-013-9892-8. |
[34] |
E. Zermelo, Über die Navigation in der Luft als Problem der Variationsrechnung, Jahresber. Deutsch. Math.-Verein., 89 (1930), 44-48. Google Scholar |
[35] |
E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung, ZAMM-Z. Angew. Math. Me., 11 (1931), 114-124. Google Scholar |
[36] |
W. Ziller,
Geometry of the Katok examples, Ergodic Theory Dyn. Syst., 3 (1983), 135-157.
doi: 10.1017/S0143385700001851. |
show all references
References:
[1] |
The International Maritime Organization (IMO), COLREG: Convention on the International Regulations for Preventing Collisions at Sea, London, United Kingdom, 2004. Google Scholar |
[2] |
N. Aldea and P. Kopacz,
Generalized Zermelo navigation on Hermitian manifolds under mild wind, Diff. Geom. Appl., 54 (2017), 325-343.
doi: 10.1016/j.difgeo.2017.05.007. |
[3] |
N. Aldea and P. Kopacz,
Generalized Zermelo navigation on Hermitian manifolds with a critical wind, Results Math., 72 (2017), 2165-2180.
doi: 10.1007/s00025-017-0757-6. |
[4] |
K. J. Arrow, On the use of winds in flight planning, J. Meteor., 6 (1949), 150-159. Google Scholar |
[5] |
D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry (eds. D. Bao et al.), Math. Sci. Res. Inst. Publ., 50 (2004), Cambridge Univ. Press, 197-259. |
[6] |
D. Bao, C. Robles and Z. Shen,
Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435.
|
[7] |
D. C. Brody, G. W. Gibbons and D. M. Meier, Time-optimal navigation through quantum wind,
New J. Phys., 17 (2015), 033048, 8pp.
doi: 10.1088/1367-2630/17/3/033048. |
[8] |
D. C. Brody, G. W. Gibbons and D. M. Meier,
A Riemannian approach to Randers geodesics, J. Geom. Phys., 106 (2016), 98-101.
doi: 10.1016/j.geomphys.2016.03.019. |
[9] |
D. C. Brody and D. M. Meier, Solution to the quantum Zermelo navigation problem, Phys. Rev. Lett., 114 (2015), 100502. Google Scholar |
[10] |
E. Caponio, M. A. Javaloyes and M. Sànchez, Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes, preprint, arXiv: 1407.5494. Google Scholar |
[11] |
C. Carathéodory,
Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965. |
[12] |
S.-S. Chern and Z. Shen,
Riemann-Finsler Geometry, Nankai tracts in mathematics, World Scientific, River Edge (N. J.), London, Singapore, 2005.
doi: 10.1142/5263. |
[13] |
M. A. Earle, Sphere to spheroid comparisons, J. Navigation, 59 (2006), 491-496. Google Scholar |
[14] |
A. de Mira Fernandes, Sul problema brachistocrono di Zermelo, Rendiconti della R. Acc. dei Lincei, XV (1932), 47-52. Google Scholar |
[15] |
C. A. R. Herdeiro, Mira Fernandes and a generalised Zermelo problem: Purely geometric
formulations, Bol. Soc. Port. Mat., 2010, Special Issue. Publication date estimated, 179-191. |
[16] |
M. R. Jardin, Toward Real-Time en Route Air Traffic Control Optimization, Ph. D thesis, Stanford University, 2003. Google Scholar |
[17] |
A. B. Katok,
Ergodic perturbations of degenerate integrable Hamiltonian systems, Math. USSR Izv., 37 (1973), 539-576.
|
[18] |
P. Kopacz, On generalization of Zermelo navigation problem on Riemannian manifolds, preprint, arXiv: 1604.06487. Google Scholar |
[19] |
P. Kopacz,
Application of planar Randers geodesics with river-type perturbation in search models, Appl. Math. Model., 49 (2017), 531-553.
doi: 10.1016/j.apm.2017.05.007. |
[20] |
P. Kopacz,
A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation, An. Sti. U. Ovid. Co.-Mat., 25 (2017), 107-123.
|
[21] |
P. Kopacz,
Application of codimension one foliation in Zermelo's problem on Riemannian manifolds, Diff. Geom. Appl., 35 (2014), 334-349.
doi: 10.1016/j.difgeo.2014.04.007. |
[22] |
P. Kopacz, Proposal on a Riemannian approach to path modeling in a navigational decision support system, in Activities of Transport Telematics. TST 2013. Communications in Computer and Information Science 395 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2013), 294-302. Google Scholar |
[23] |
T. Levi-Civita, Über Zermelo's Luftfahrtproblem, ZAMM-Z. Angew. Math. Me., 11 (1931), 314-322. Google Scholar |
[24] |
B. Manià,
Sopra un problema di navigazione di Zermelo, Math. Ann., 133 (1937), 584-599.
doi: 10.1007/BF01571651. |
[25] |
A. Pallikaris and G. Latsas, New Algorithm for Great Elliptic Sailing (GES), J. Navigation, 62 (2012), 493-507. Google Scholar |
[26] |
A. Pressley,
Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, London, 2010.
doi: 10.1007/978-1-84882-891-9. |
[27] |
C. Robles,
Geodesics in Randers spaces of constant curvature, T. Am. Math. Soc., 359 (2007), 1633-1651.
doi: 10.1090/S0002-9947-06-04051-7. |
[28] |
B. Russell and S. Stepney, Zermelo navigation in the quantum brachistochrone,
J. Phys. A - Math. Theor., 48 (2015), 115303, 29pp.
doi: 10.1088/1751-8113/48/11/115303. |
[29] |
Z. Shen,
Finsler Metrics with ${\bf{K}} = 0$ and ${\bf{S}} = 0$, Can. J. Math., 55 (2003), 112-132.
doi: 10.4153/CJM-2003-005-6. |
[30] |
W.-K. Tseng, M. A. Earle and J.-L. Guo, Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing, J. Navigation, 65 (2012), 549-559. Google Scholar |
[31] |
A. Weintrit and P. Kopacz, Computational algorithms implemented in marine navigation electronic systems, in Telematics in the Transport Environment. TST 2012. Communications in Computer and Information Science 329 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2012), 148-158. Google Scholar |
[32] |
A. Weintrit and P. Kopacz, A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General, in Methods and Algorithms in Navigation (eds. A. Weintrit and T. Neumann), CRC Press, (2011), 123-132. Google Scholar |
[33] |
R. Yoshikawa and S. V. Sabau,
Kropina metrics and Zermelo navigation on Riemannian manifolds, Geom. Dedicata, 171 (2014), 119-148.
doi: 10.1007/s10711-013-9892-8. |
[34] |
E. Zermelo, Über die Navigation in der Luft als Problem der Variationsrechnung, Jahresber. Deutsch. Math.-Verein., 89 (1930), 44-48. Google Scholar |
[35] |
E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung, ZAMM-Z. Angew. Math. Me., 11 (1931), 114-124. Google Scholar |
[36] |
W. Ziller,
Geometry of the Katok examples, Ergodic Theory Dyn. Syst., 3 (1983), 135-157.
doi: 10.1017/S0143385700001851. |




























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