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Research Article | Open Access

A natural 4th-order generalization of the geodesic problem

CMUC, Department of Mathematics, University of Coimbra, Coimbra 3000–143, Portugal
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Abstract

We propose a fourth-order extension of the geodesic problem arising in the continuity of the study of Riemannian cubics. We consider the variational problem in a Riemannian manifold and derive the Euler-Lagrange equation. For the special situation of Lie groups, we use Euler-Poincaré reduction to obtain the reduced equation on the Lie algebra.

Keywords

geodesicsRiemannian cubicsEuler-Lagrange equationsEuler-Poincaré equations
CLC number: 53C22, 58E10, 70H50

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Electronic Research Archive
Volume 32 Issue 5,
May 2024
Pages 3396-3412
DOI: 10.3934/era.2024157

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Camarinha M. A natural 4th-order generalization of the geodesic problem. Electronic Research Archive, 2024, 32(5): 3396-3412. https://doi.org/10.3934/era.2024157

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Received: 29 February 2024
Revised: 01 May 2024
Accepted: 16 May 2024
Published: 15 May 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)