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An Intrinsic Version of the k -Harmonic Equation

Author

Listed:
  • Lígia Abrunheiro

    (Aveiro Institute of Accounting and Administration of the University of Aveiro (ISCA-UA), 3810-500 Aveiro, Portugal
    Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
    These authors contributed equally to this work.)

  • Margarida Camarinha

    (CMUC, University of Coimbra, Department of Mathematics, 3000-143 Coimbra, Portugal
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Lígia Abrunheiro & Margarida Camarinha, 2023. "An Intrinsic Version of the k -Harmonic Equation," Mathematics, MDPI, vol. 11(17), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3628-:d:1222637
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