{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T00:05:30Z","timestamp":1775261130939,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,11,1]],"date-time":"2018-11-01T00:00:00Z","timestamp":1541030400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2019,9]]},"DOI":"10.1007\/s10957-018-1424-0","type":"journal-article","created":{"date-parts":[[2018,11,1]],"date-time":"2018-11-01T06:47:44Z","timestamp":1541054864000},"page":"965-983","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres"],"prefix":"10.1007","volume":"182","author":[{"given":"Lu\u00eds","family":"Machado","sequence":"first","affiliation":[]},{"given":"L\u00edgia","family":"Abrunheiro","sequence":"additional","affiliation":[]},{"given":"Nat\u00e1lia","family":"Martins","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,11,1]]},"reference":[{"key":"1424_CR1","volume-title":"Variational Methods in Nonconservative Phenomena. Mathematics in Science and Engineering","author":"BD Vujanovic","year":"1989","unstructured":"Vujanovic, B.D., Jones, S.E.: Variational Methods in Nonconservative Phenomena. Mathematics in Science and Engineering, vol. 182. Academic Press Inc., Cambridge (1989)"},{"key":"1424_CR2","unstructured":"Herglotz, G.: Ber\u00fchrungstransformationen. Lectures at the University of G\u00f6ttingen, G\u00f6ttingen (1930)"},{"issue":"2","key":"1424_CR3","doi-asserted-by":"publisher","first-page":"261","DOI":"10.12775\/TMNA.2002.036","volume":"20","author":"B Georgieva","year":"2002","unstructured":"Georgieva, B., Guenther, R.: First Noether-type theorem for the generalized variational principle of Herglotz. Topol. Methods Nonlinear Anal. 20(2), 261\u2013273 (2002)","journal-title":"Topol. Methods Nonlinear Anal."},{"issue":"2","key":"1424_CR4","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1137\/1038042","volume":"38","author":"R Guenther","year":"1996","unstructured":"Guenther, R., Gottsch, J.A., Kramer, D.B.: The Herglotz algorithm for constructing canonical transformations. SIAM Rev. 38(2), 287\u2013293 (1996)","journal-title":"SIAM Rev."},{"key":"1424_CR5","volume-title":"The Herglotz Lectures on Contact Transformations and Hamiltonian Systems. Lecture Notes in Nonlinear Analysis","author":"R Guenther","year":"1996","unstructured":"Guenther, R., Guenther, C.M., Gottsch, J.A.: The Herglotz Lectures on Contact Transformations and Hamiltonian Systems. Lecture Notes in Nonlinear Analysis, vol. 1. Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Tor\u00fan (1996)"},{"key":"1424_CR6","first-page":"113","volume":"100","author":"B Georgieva","year":"2010","unstructured":"Georgieva, B.: Symmetries of the Herglotz variational principle in the case of one independent variable. Ann. Sofia Univ. Fac. Math. Inf. 100, 113\u2013122 (2010)","journal-title":"Ann. Sofia Univ. Fac. Math. Inf."},{"issue":"2","key":"1424_CR7","doi-asserted-by":"publisher","first-page":"307","DOI":"10.12775\/TMNA.2005.034","volume":"26","author":"B Georgieva","year":"2005","unstructured":"Georgieva, B., Guenther, R.: Second Noether-type theorem for the generalized variational principle of Herglotz. Topol. Methods Nonlinear Anal. 26(2), 307\u2013314 (2005)","journal-title":"Topol. Methods Nonlinear Anal."},{"issue":"9","key":"1424_CR8","doi-asserted-by":"publisher","first-page":"3911","DOI":"10.1063\/1.1597419","volume":"44","author":"B Georgieva","year":"2003","unstructured":"Georgieva, B., Guenther, R., Bodurov, T.: Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem. J. Math. Phys. 44(9), 3911\u20133927 (2003)","journal-title":"J. Math. Phys."},{"issue":"9","key":"1424_CR9","doi-asserted-by":"publisher","first-page":"4593","DOI":"10.3934\/dcds.2015.35.4593","volume":"35","author":"SPS Santos","year":"2015","unstructured":"Santos, S.P.S., Martins, N., Torres, D.F.M.: Variational problems of Herglotz type with time delay: Dubois\u2013Reymond condition and Noether\u2019s first theorem. Discret. Contin. Dyn. Syst. Ser. A 35(9), 4593\u20134610 (2015)","journal-title":"Discret. Contin. Dyn. Syst. Ser. A"},{"key":"1424_CR10","first-page":"107","volume":"449","author":"SPS Santos","year":"2015","unstructured":"Santos, S.P.S., Martins, N., Torres, D.F.M.: An optimal control approach to Herglotz variational problems. Optimization in the natural sciences. Commun. Comput. Inf. Sci. (CCIS) 449, 107\u2013117 (2015)","journal-title":"Commun. Comput. Inf. Sci. (CCIS)"},{"key":"1424_CR11","doi-asserted-by":"crossref","unstructured":"Santos, S.P.S., Martins, N., Torres, D.F.M.: Noether\u2019s theorem for higher-order variational problems of Herglotz type. In: 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, vol. 2015, pp. 990\u2013999 (2015)","DOI":"10.3934\/proc.2015.990"},{"issue":"2","key":"1424_CR12","first-page":"291","volume":"1","author":"SPS Santos","year":"2016","unstructured":"Santos, S.P.S., Martins, N., Torres, D.F.M.: Higher-order variational problems of Herglotz type with time delay. Pure Appl. Funct. Anal. 1(2), 291\u2013307 (2016)","journal-title":"Pure Appl. Funct. Anal."},{"issue":"1","key":"1424_CR13","doi-asserted-by":"crossref","first-page":"91","DOI":"10.3934\/dcdss.2018006","volume":"11","author":"SPS Santos","year":"2018","unstructured":"Santos, S.P.S., Martins, N., Torres, D.F.M.: Noether currents for higher-order variational problems of Herglotz type with time delay. Discret. Contin. Dyn. Syst. Ser. S 11(1), 91\u2013102 (2018)","journal-title":"Discret. Contin. Dyn. Syst. Ser. S"},{"issue":"4","key":"1424_CR14","doi-asserted-by":"publisher","first-page":"1481","DOI":"10.1007\/s00707-016-1758-3","volume":"228","author":"Y Zhang","year":"2017","unstructured":"Zhang, Y.: Variational problem of Herglotz type for Birkhoffian system and its Noether\u2019s theorems. Acta Mech. 228(4), 1481\u20131492 (2017)","journal-title":"Acta Mech."},{"issue":"3","key":"1424_CR15","doi-asserted-by":"publisher","first-page":"887","DOI":"10.1007\/s10773-017-3621-2","volume":"57","author":"X Tian","year":"2018","unstructured":"Tian, X., Zhang, Y.: Noether\u2019s theorem and its inverse of Birkhoffian system in event space based on Herglotz variational problem. Int. J. Theor. Phys. 57(3), 887\u2013897 (2018)","journal-title":"Int. J. Theor. Phys."},{"key":"1424_CR16","doi-asserted-by":"crossref","DOI":"10.1090\/coll\/009","volume-title":"Dynamical Systems","author":"GD Birkhoff","year":"1927","unstructured":"Birkhoff, G.D.: Dynamical Systems. AMS College Publication, Providence (1927)"},{"key":"1424_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-86760-6","volume-title":"Foundations of Theoretical Mechanics II","author":"RM Santilli","year":"1983","unstructured":"Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, Berlin (1983)"},{"issue":"8","key":"1424_CR18","doi-asserted-by":"publisher","first-page":"2367","DOI":"10.3934\/dcdsb.2014.19.2367","volume":"19","author":"R Almeida","year":"2014","unstructured":"Almeida, R., Malinowska, A.B.: Fractional variational principle of Herglotz. Discret. Contin. Dyn. Syst. Ser. B 19(8), 2367\u20132381 (2014)","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"key":"1424_CR19","doi-asserted-by":"publisher","first-page":"1890","DOI":"10.1103\/PhysRevE.53.1890","volume":"53","author":"F Riewe","year":"1996","unstructured":"Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890\u20131899 (1996)","journal-title":"Phys. Rev. E"},{"key":"1424_CR20","doi-asserted-by":"publisher","first-page":"3582","DOI":"10.1103\/PhysRevE.55.3581","volume":"55","author":"F Riewe","year":"1997","unstructured":"Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3582\u20133592 (1997)","journal-title":"Phys. Rev. E"},{"key":"1424_CR21","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/s11071-004-3764-6","volume":"38","author":"OP Agrawal","year":"2004","unstructured":"Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323\u2013337 (2004)","journal-title":"Nonlinear Dyn."},{"issue":"9\u201310","key":"1424_CR22","doi-asserted-by":"publisher","first-page":"1269","DOI":"10.1177\/1077546307077467","volume":"13","author":"OP Agrawal","year":"2007","unstructured":"Agrawal, O.P., Baleanu, D.: Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13(9\u201310), 1269\u20131281 (2007)","journal-title":"J. Vib. Control"},{"key":"1424_CR23","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1007\/s11071-010-9748-9","volume":"62","author":"F Jarad","year":"2010","unstructured":"Jarad, F., Abdeljawad, T., Baleanu, D.: Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn. 62, 609\u2013614 (2010)","journal-title":"Nonlinear Dyn."},{"key":"1424_CR24","doi-asserted-by":"crossref","first-page":"9234","DOI":"10.1016\/j.amc.2012.02.080","volume":"218","author":"F Jarad","year":"2012","unstructured":"Jarad, F., Abdeljawad, T., Baleanu, D.: Higher order fractional variational optimal control problems with delayed arguments. Appl. Math. Comput. 218, 9234\u20139240 (2012)","journal-title":"Appl. Math. Comput."},{"key":"1424_CR25","first-page":"1","volume":"69","author":"GM Bahaa","year":"2017","unstructured":"Bahaa, G.M.: Fractional optimal control problem for differential system with delay argument. Adv. Differ. Equ. 69, 1\u201319 (2017)","journal-title":"Adv. Differ. Equ."},{"issue":"1","key":"1424_CR26","first-page":"12","volume":"7","author":"L Abrunheiro","year":"2016","unstructured":"Abrunheiro, L., Machado, L., Martins, N.: The Herglotz variational problem on spheres and its optimal control approach. J. Math. Anal. 7(1), 12\u201322 (2016)","journal-title":"J. Math. Anal."},{"key":"1424_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2063-1","volume-title":"Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics","author":"VI Arnold","year":"1989","unstructured":"Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60, 2nd edn. Springer, Berlin (1989)","edition":"2"},{"key":"1424_CR28","doi-asserted-by":"crossref","unstructured":"Bloch, A.M, Crouch, P.E.: Reduction of Euler\u2013Lagrange problems for constrained variational problems and relation with optimal control problems. In: Proceedings of the 33rd Conference on Decision and Control, Orlando, FL, pp. 2584\u20132590 (1994)","DOI":"10.1109\/CDC.1994.411534"},{"key":"1424_CR29","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1137\/S036301299223533X","volume":"33","author":"AM Bloch","year":"1995","unstructured":"Bloch, A.M., Crouch, P.E.: Nonholonomic control systems on Riemannian manifolds. SIAM J. Control Optim. 33, 126\u2013148 (1995)","journal-title":"SIAM J. Control Optim."},{"key":"1424_CR30","doi-asserted-by":"crossref","unstructured":"Bloch, A.M., Crouch, P.E.: On the equivalence of higher order variational problems and optimal control problems. In: Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, pp. 1648\u20131653 (1996)","DOI":"10.1109\/CDC.1996.572780"},{"key":"1424_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2201-7","volume-title":"Riemannian Geometry. Mathematics: Theory and Applications","author":"MP Carmo do","year":"1992","unstructured":"do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory and Applications. Birk\u00e4user, Boston (1992)"},{"key":"1424_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-21298-7","volume-title":"Riemannian Geometry and Geometric Analysis. Universitext","author":"J Jost","year":"2011","unstructured":"Jost, J.: Riemannian Geometry and Geometric Analysis. Universitext, 6th edn. Springer, Berlin (2011)","edition":"6"},{"key":"1424_CR33","volume-title":"Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives. North-Holland Mathematics Studies, Nachbin, L. (ed.)","author":"M Le\u00f3n de","year":"1985","unstructured":"de Le\u00f3n, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives. North-Holland Mathematics Studies, Nachbin, L. (ed.), vol. 112. Elsiever, Amsterdam (1985)"},{"key":"1424_CR34","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1007\/s10013-013-0048-9","volume":"42","author":"SPS Santos","year":"2014","unstructured":"Santos, S.P.S., Martins, N., Torres, D.F.M.: Higher-order variational problems of Herglotz type. Vietnam J. Math. 42, 409\u2013419 (2014)","journal-title":"Vietnam J. Math."},{"issue":"2","key":"1424_CR35","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1023\/A:1023289721398","volume":"9","author":"B Bonnard","year":"2003","unstructured":"Bonnard, B., Faubourg, L., Launay, G., Tr\u00e9lat, E.: Optimal control with state constraints and the space shuttle re-entry problem. J. Dyn. Control Syst. 9(2), 155\u2013199 (2003)","journal-title":"J. Dyn. Control Syst."},{"key":"1424_CR36","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/0022-247X(71)90219-8","volume":"35","author":"DH Jacobson","year":"1971","unstructured":"Jacobson, D.H., Lee, M.M.: New necessary conditions of optimality for control problems with state-variable inequality constraints. J. Math. Anal. Appl. 35, 255\u2013284 (1971)","journal-title":"J. Math. Anal. Appl."},{"key":"1424_CR37","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1093\/imamci\/6.4.465","volume":"6","author":"L Noakes","year":"1989","unstructured":"Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inf. 6, 465\u2013473 (1989)","journal-title":"IMA J. Math. Control Inf."},{"key":"1424_CR38","doi-asserted-by":"crossref","unstructured":"Abrunheiro, L. Camarinha, M., Clemente-Gallardo, J.: Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration. In: The Cape Verde International Days on Mathematics 2013, Praia, Cape Verde, Conference Papers in Mathematics, pp. 1\u20139. Hindawi Publishing Corporation (2013)","DOI":"10.1155\/2013\/243621"},{"key":"1424_CR39","volume-title":"An Introduction to Nonlinear Oscillations","author":"RE Mickens","year":"1981","unstructured":"Mickens, R.E.: An Introduction to Nonlinear Oscillations. Cambridge University Press, Cambridge (1981)"},{"issue":"2","key":"1424_CR40","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1119\/1.14703","volume":"54","author":"RA Nelson","year":"1986","unstructured":"Nelson, R.A., Olsson, M.G.: The pendulum: Rich physics from a simple system. Am. J. Phys. 54(2), 112\u2013121 (1986)","journal-title":"Am. J. Phys."},{"issue":"2","key":"1424_CR41","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1007\/PL00009863","volume":"13","author":"M Camarinha","year":"2000","unstructured":"Camarinha, M., Silva Leite, F., Crouch, P.: Elastic curves as solutions of Riemannian and sub-Riemannian control problems. Math. Control Signal Syst. 13(2), 140\u2013155 (2000)","journal-title":"Math. Control Signal Syst."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-018-1424-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-018-1424-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-018-1424-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T22:51:10Z","timestamp":1775256670000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-018-1424-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,1]]},"references-count":41,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,9]]}},"alternative-id":["1424"],"URL":"https:\/\/doi.org\/10.1007\/s10957-018-1424-0","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,1]]},"assertion":[{"value":"25 November 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 October 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 November 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}