An Intrinsic Version of the k -Harmonic Equation

Lígia Abrunheiro and Margarida Camarinha ()
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Lígia Abrunheiro: Aveiro Institute of Accounting and Administration of the University of Aveiro (ISCA-UA), 3810-500 Aveiro, Portugal
Margarida Camarinha: CMUC, University of Coimbra, Department of Mathematics, 3000-143 Coimbra, Portugal

Mathematics, 2023, vol. 11, issue 17, 1-19

Abstract: The notion of k -harmonic curves is associated with the k th-order variational problem defined by the k -energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the k th-order tangent bundle T k M to the cotangent bundle T * T k − 1 M . The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves.

Keywords: k -harmonic curves; Riemannian manifolds; Lagrangian and Hamiltonian formalism; Legendre transformation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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