{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:25:58Z","timestamp":1760149558089,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2023,8,22]],"date-time":"2023-08-22T00:00:00Z","timestamp":1692662400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Portuguese Foundation for Science and Technology (FCT)","award":["UIDB\/04106\/2020","UIDB\/00324\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020","UIDB\/00324\/2020"]}]},{"name":"Centre for Mathematics of the University of Coimbra","award":["UIDB\/04106\/2020","UIDB\/00324\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020","UIDB\/00324\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>The notion of k-harmonic curves is associated with the kth-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 kth-order tangent bundle TkM to the cotangent bundle T*Tk\u22121M. The intrinsic version of the Euler\u2013Lagrange 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.<\/jats:p>","DOI":"10.3390\/math11173628","type":"journal-article","created":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T08:01:21Z","timestamp":1692777681000},"page":"3628","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Intrinsic Version of the k-Harmonic Equation"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5950-9475","authenticated-orcid":false,"given":"L\u00edgia","family":"Abrunheiro","sequence":"first","affiliation":[{"name":"Aveiro Institute of Accounting and Administration of the University of Aveiro (ISCA-UA), 3810-500 Aveiro, Portugal"},{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4587-7861","authenticated-orcid":false,"given":"Margarida","family":"Camarinha","sequence":"additional","affiliation":[{"name":"CMUC, University of Coimbra, Department of Mathematics, 3000-143 Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.jde.2020.11.046","article-title":"A structure theorem for polyharmonic maps between Riemannian manifolds","volume":"273","author":"Branding","year":"2021","journal-title":"J. Differ. Equ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"013508","DOI":"10.1063\/1.2830433","article-title":"Unit speed stationary points of the acceleration","volume":"49","author":"Arroyo","year":"2008","journal-title":"J. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s00009-006-0090-x","article-title":"The Euler\u2013Lagrange method for biharmonic curves","volume":"3","author":"Caddeo","year":"2006","journal-title":"Mediterr. J. Math."},{"key":"ref_4","first-page":"1","article-title":"A short survey on biharmonic maps between Riemannian manifolds","volume":"47","author":"Montaldo","year":"2006","journal-title":"Rev. Uni\u00f3n Mat. Argent."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1007\/s00009-021-01837-y","article-title":"Triharmonic Curves in 3-Dimensional Homogeneous Spaces","volume":"18","author":"Montaldo","year":"2021","journal-title":"Mediterr. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"651","DOI":"10.3390\/sym7020651","article-title":"Harmonic maps and biharmonic maps","volume":"7","author":"Urakawa","year":"2015","journal-title":"Symmetry"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1093\/imamci\/6.4.465","article-title":"Cubic splines on curved spaces","volume":"6","author":"Noakes","year":"1989","journal-title":"IMA J. Math. Control Inf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF02254638","article-title":"The dynamic interpolation problem: On Riemannian manifolds, Lie groups and symmetric spaces","volume":"1","author":"Crouch","year":"1995","journal-title":"J. Dyn. Control Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1093\/imamci\/12.4.399","article-title":"Splines of class Ck on non-euclidean spaces","volume":"12","author":"Camarinha","year":"1995","journal-title":"IMA J. Math. Control Inf."},{"key":"ref_10","unstructured":"Camarinha, M., Crouch, P., and Silva Leite, F. (2023, July 19). High-order splines on Riemannian manifolds, Tr. Mat. Inst. Im. V. A. Steklova, Available online: https:\/\/www.mathnet.ru\/eng\/tm4324."},{"key":"ref_11","first-page":"13","article-title":"Hamiltonian structure of generalized cubic polynomials","volume":"Volune 33","author":"Camarinha","year":"2000","journal-title":"N. E. Leonard and R. Ortega, editors, Lagrangian and Hamiltonian Methods for Nonlinear Control 2000: A Proceedings Volume from the IFACWorkshop, Princeton, NJ, USA, 16\u201318 March 2000"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1051\/cocv:2004018","article-title":"Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric","volume":"10","author":"Altafini","year":"2004","journal-title":"J. ESAIM Control Optim. Calc. Var."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1093\/imamci\/dni069","article-title":"Quadratures and cubics in SO(3) and SO(1,2)","volume":"23","author":"Noakes","year":"2006","journal-title":"IMA J. Math. Control Inf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"355203","DOI":"10.1088\/1751-8113\/44\/35\/355203","article-title":"Cubic polynomials on Lie groups: Reduction of the Hamiltonian system","volume":"44","author":"Abrunheiro","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s00220-011-1313-y","article-title":"Invariant Higher-Order Variational Problems","volume":"309","author":"Holm","year":"2012","journal-title":"Commun. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"451","DOI":"10.3934\/jgm.2014.6.451","article-title":"Higher-order variational problems on Lie groups and optimal control applications","volume":"6","author":"Colombo","year":"2014","journal-title":"J. Geom. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"257","DOI":"10.3934\/jgm.2017011","article-title":"About simple variational splines from the Hamiltonian viewpoint","volume":"9","author":"Balseiro","year":"2017","journal-title":"J. Geom. Mech."},{"key":"ref_18","unstructured":"de Le\u00f3n, M., and Rodrigues, P.R. (1985). Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives, North-Holland, Elsiever. North-Holland Math. Studies."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Lee, L.M. (1997). Riemannian Manifolds: An Introduction to Curvature, Springer.","DOI":"10.1007\/0-387-22726-1_7"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Abrunheiro, L., Camarinha, M., and Clemente-Gallardo, J. (2013). Conference Papers in Mathematics, Hindawi Publishing Corporation.","DOI":"10.1155\/2013\/243621"}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/17\/3628\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:39:49Z","timestamp":1760128789000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/17\/3628"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,22]]},"references-count":20,"journal-issue":{"issue":"17","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["math11173628"],"URL":"https:\/\/doi.org\/10.3390\/math11173628","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2023,8,22]]}}}