On Funk's parabolas

Authors

DOI:

https://doi.org/10.35819/remat2024v10i1id6680

Keywords:

Finsler metric, Funk metric, navigation problem, Funk's parabolas

Abstract

This study aims to analyze parabolas in the two dimensional unit disk equipped with a Funk metric. The analysis leads to four types of parabolas are obtained, due to the non-reversibility of the Funk metric. Each one with applications to physics in the Zermelo navigation problem. In addition, we identify that two of the four parabolas obtained are in well known Euclidian conics, and the remaining two are characterized by irreducible quartics.

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Author Biographies

Newton Mayer Solórzano Chávez, Universidade Federal da Integração Latino-Americana (UNILA), Foz do Iguaçu, PR, Brasil

http://lattes.cnpq.br/5434324704126162

Junior Rodrigues Moyses, Universidade Federal de Goiás (UFG), Goiânia, GO, Brasil

http://lattes.cnpq.br/2207888598903867

Víctor Arturo Martínez León, Universidade Federal da Integração Latino-Americana (UNILA), Foz do Iguaçu, PR, Brasil

http://lattes.cnpq.br/9966158817422161

References

BOULOS, P.; CAMARGO, I. Geometria Analítica: um tratamento vetorial. 3. ed. São Paulo: Prentice Hall, 2005.

CARMO, M. P. do. Geometria Riemanniana. 6. ed. Rio de Janeiro: IMPA, 2019.

CHÁVEZ, N. M. S.; LEÓN, V. A. M.; SOSA, L. G. Q.; MOYSES, J. R. Um problema de navegação de Zermelo: Métrica de Funk. REMAT: Revista Eletrônica da Matemática, Bento Gonçalves, v. 7, n. 1, p. e3010, 29 mar. 2021. DOI: https://doi.org/10.35819/remat2021v7i1id4574.

CHENG, X.; SHEN, Z. Finsler Geometry: An approach via Randers spaces. Beijing-Heidelberg: Science Press Beijing-Springer, 2012.

CHERN, S. S.; SHEN, Z. Riemannian-Finsler geometry. Singapore: World Scientific, 2005.

SHEN, Z. Lectures on Finsler Geometry. Singapore: World Scientific, 2001.

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Published

2024-01-31

How to Cite

CHÁVEZ, N. M. S.; MOYSES, J. R.; LEÓN, V. A. M. On Funk’s parabolas. REMAT: Revista Eletrônica da Matemática, Bento Gonçalves, RS, v. 10, n. 1, p. e3001, 2024. DOI: 10.35819/remat2024v10i1id6680. Disponível em: https://periodicos.ifrs.edu.br/index.php/REMAT/article/view/6680. Acesso em: 11 may. 2024.

Issue

Vol. 10 No. 1 (2024)

Section

Mathematics

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Copyright (c) 2024 REMAT: Revista Eletrônica da Matemática

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