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

Piotr Kopacz

doi:10.1515/auom-2017-0039

.blurhash-client-img { display: none !important; }

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

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 25 (2017): Issue 3 (December 2017)

By: Piotr Kopacz

Open Access

|Mar 2018

Abstract

We generalize the Zermelo navigation on Riemannian manifolds (M; h), admitting a space dependence of a ship's speed 0 < |u(x)|_h ≤ 1 in the presence of a perturbation W̃ determined by a strong (critical) velocity vector field satisfying |W̃ (x)|_h = |u(x)|_h, with application of Finsler metric of Kropina type.

References

[1] N. Aldea. Complex Finsler spaces with Kropina metric. Bull. Transilvania Univ. Braşov (Proc. Conference RIGA 2007) 14 (49s.) (2007) 1-10.
Search in Google Scholar Back to article
[2] N. Aldea, P. Kopacz. Generalized Zermelo navigation on Hermitian manifolds under mild wind. Diff. Geom. Appl. (2017), in press.10.1016/j.difgeo.2017.05.007
Open DOI Search in Google Scholar Back to article
[3] N. Aldea, P. Kopacz. Generalized Zermelo navigation on Hermitian manifolds with a critical wind. Results Math. (2017) 72(4): 2165-2180, doi:10.1007/s00025-017-0757-6.
Open DOI Search in Google Scholar Back to article
[4] K. J. Arrow. On the use of winds in fight planning. J. Meteor. 6 (1949) 150-159.10.1175/1520-0469(1949)006<;0150:OTUOWI>2.0.CO;2
Search in Google Scholar Back to article
[5] D. Bao, C. Robles, and Z. Shen. Zermelo navigation on Riemannian manifolds. J. Di-. Geom. 66(3) (2004) 377-435.10.4310/jdg/1098137838
Search in Google Scholar Back to article
[6] D. C. Brody, G. W. Gibbons, and D. M. Meier. Time-optimal navigation through quantum wind. New J. Phys. 17 (033048) (2015).10.1088/1367-2630/17/3/033048
Search in Google Scholar Back to article
[7] D. C. Brody, G. W. Gibbons, and D. M. Meier. A Riemannian approach to Randers geodesics. J. Geom. Phys. 106 (2016) 98-101.10.1016/j.geomphys.2016.03.019
Search in Google Scholar Back to article
[8] D. C. Brody and D. M. Meier. Solution to the quantum Zermelo navigation problem. Phys. Rev. Lett. 114 (100502) (2015).10.1103/PhysRevLett.114.10050225815915
Search in Google Scholar Back to article
[9] E. Caponio, M. A. Javaloyes, and M. Sánchez. Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes. arXiv:1407.5494, 2015.
Search in Google Scholar Back to article
[10] C. Carathéodory. Calculus of Variations and Partial Differential Equations of the First Order . AMS, Chelsea Publishing, 1935 (reprint 2008).
Search in Google Scholar Back to article
[11] S.-S. Chern and Z. Shen. Riemann-Finsler geometry. Nankai Tracts in Mathematics. World Scientific, River Edge (N.J.), London, Singapore, 2005.10.1142/5263
Open DOI Search in Google Scholar Back to article
[12] A. De Mira Fernandes. Sul problema brachistocrono di Zermelo. Rendiconti della R. Acc. dei Lincei XV (1932) 47-52.
Search in Google Scholar Back to article
[13] C. A. R. Herdeiro. Mira Fernandes and a generalised Zermelo problem: purely geometric formulations. Bol. Soc. Port. Mat., Special Issue. Proc. of "Mira Fernandes and his age - An historical conference in honour of Aureliano de Mira Fernandes (1884-1958)"', Instituto Superior Técnico, Lisboa, 179-191, June 2009 (Eds. L. Saraiva and J. T. Pinto), Lisboa, 1-13, 2010.
Search in Google Scholar Back to article
[14] P. Kopacz. Application of planar Randers geodesics with river-type perturbation in search models. Appl. Math. Model. 49 (2017) 531-553.10.1016/j.apm.2017.05.007
Search in Google Scholar Back to article
[15] P. Kopacz. On generalization of Zermelo navigation problem on Riemannian manifolds. arXiv:1604.06487 [math.DG], 2016.10.1515/auom-2017-0039
Search in Google Scholar Back to article
[16] P. Kopacz. A note on time-optimal paths on perturbed spheroid. arXiv:1602.00167 [math.DG], 2016.
Search in Google Scholar Back to article
[17] T. Levi-Civita. Über Zermelo's Luftfahrtproblem. ZAMM-Z. Angew. Math. Me. 11(4) (1931) 314-322.10.1002/zamm.19310110404
Search in Google Scholar Back to article
[18] B. Manià. Sopra un problema di navigazione di Zermelo. Math. Ann. 133 (3) (1937) 584-599.10.1007/BF01571651
Search in Google Scholar Back to article
[19] V. S. Matveev, H.-B. Rademacher, M. Troyanov, and A. Zeghib. Finsler conformal Lichnerowicz-Obata conjecture. Ann. I. Fourier 59 (3) (2009) 937-949.10.5802/aif.2452
Search in Google Scholar Back to article
[20] M. Ra-e-Rad. Weakly conformal Finsler geometry. Math. Nachr. 287 (14-15) (2014) 1745-1755.10.1002/mana.201300099
Search in Google Scholar Back to article
[21] B. Russell and S. Stepney. Zermelo navigation and a speed limit to quantum information processing. Phys. Rev. A 90 (012303) (2014).10.1103/PhysRevA.90.012303
Search in Google Scholar Back to article
[22] B. Russell and S. Stepney. Zermelo navigation in the quantum brachistochrone. J. Phys. A - Math. Theor. 48 (115303) (2015).10.1088/1751-8113/48/11/115303
Search in Google Scholar Back to article
[23] R. Yoshikawa and S. V. Sabau. Kropina metrics and Zermelo navigation on Riemannian manifolds. Geom. Dedicata 171 (1) (2013) 119-148.10.1007/s10711-013-9892-8
Search in Google Scholar Back to article
[24] E. Zermelo. Über die Navigation in der Luft als Problem der Variationsrechnung. Jahresber. Deutsch. Math.-Verein. 89 (1930) 44-48.
Search in Google Scholar Back to article
[25] E. Zermelo. Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung. ZAMM-Z. Angew. Math. Me. 11 (1931) 114-124.10.1002/zamm.19310110205
Open DOI Search in Google Scholar Back to article