A natural 4th-order generalization of the geodesic problem

Margarida Camarinha; Margarida Camarinha

doi:10.3934/era.2024157

Electronic Research Archive

2024, Volume 32, Issue 5: 3396-3412. doi: 10.3934/era.2024157

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Research article

A natural 4th-order generalization of the geodesic problem

Margarida Camarinha ^,

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

Received: 29 February 2024 Revised: 01 May 2024 Accepted: 16 May 2024 Published: 27 May 2024
53C22, 58E10, 70H50

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.
- geodesics,
- Riemannian cubics,
- Euler-Lagrange equations,
- Euler-Poincaré equations
Citation: Margarida Camarinha. A natural 4th-order generalization of the geodesic problem[J]. Electronic Research Archive, 2024, 32(5): 3396-3412. doi: 10.3934/era.2024157

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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.

References

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