{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T22:57:04Z","timestamp":1777503424033,"version":"3.51.4"},"publisher-location":"New York, NY","reference-count":33,"publisher":"Springer New York","isbn-type":[{"value":"9781461404569","type":"print"},{"value":"9781461404576","type":"electronic"}],"license":[{"start":{"date-parts":[[2011,10,21]],"date-time":"2011-10-21T00:00:00Z","timestamp":1319155200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2011,10,21]],"date-time":"2011-10-21T00:00:00Z","timestamp":1319155200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012]]},"DOI":"10.1007\/978-1-4614-0457-6_9","type":"book-chapter","created":{"date-parts":[[2011,11,17]],"date-time":"2011-11-17T10:33:28Z","timestamp":1321526008000},"page":"109-118","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Fractional Euler\u2013Lagrange Differential Equations via Caputo Derivatives"],"prefix":"10.1007","author":[{"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[]},{"given":"Agnieszka B.","family":"Malinowska","sequence":"additional","affiliation":[]},{"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,10,21]]},"reference":[{"issue":"9\u201310","key":"9_CR1_9","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1177\/1077546307077472","volume":"13","author":"OP Agrawal","year":"2007","unstructured":"Agrawal OP (2007) Generalized Euler-Lagrange equations and transversality conditions for fvps in terms of the Caputo derivative. J Vib Contr 13(9\u201310):1217\u20131237","journal-title":"J Vib Contr"},{"key":"9_CR2_9","unstructured":"Almeida R, Malinowska AB Generalized transversality conditions in fractional calculus of variations. (Submitted)"},{"issue":"3","key":"9_CR3_9","doi-asserted-by":"crossref","first-page":"033503","DOI":"10.1063\/1.3319559","volume":"51","author":"R Almeida","year":"2010","unstructured":"Almeida R, Malinowska AB, Torres DFM (2010) A fractional calculus of variations for multiple integrals with application to vibrating string. J Math Phys 51(3):033503, 12 pp","journal-title":"J Math Phys"},{"issue":"2","key":"9_CR4_9","doi-asserted-by":"publisher","first-page":"674","DOI":"10.1016\/j.jmaa.2009.06.029","volume":"359","author":"R Almeida","year":"2009","unstructured":"Almeida R, Torres DFM (2009) H\u00f6lderian variational problems subject to integral constraints. J Math Anal Appl 359(2):674\u2013681","journal-title":"J Math Anal Appl"},{"issue":"6","key":"9_CR5_9","doi-asserted-by":"publisher","first-page":"951","DOI":"10.1177\/1077546309103268","volume":"15","author":"R Almeida","year":"2009","unstructured":"Almeida R, Torres DFM (2009) Isoperimetric problems on time scales with nabla derivatives. J Vib Contr 15(6):951\u2013958","journal-title":"J Vib Contr"},{"issue":"12","key":"9_CR6_9","doi-asserted-by":"publisher","first-page":"1816","DOI":"10.1016\/j.aml.2009.07.002","volume":"22","author":"R Almeida","year":"2009","unstructured":"Almeida R, Torres DFM (2009) Calculus of variations with fractional derivatives and fractional integrals. Appl Math Lett 22(12):1816\u20131820","journal-title":"Appl Math Lett"},{"issue":"3","key":"9_CR7_9","doi-asserted-by":"publisher","first-page":"956","DOI":"10.1016\/j.amc.2010.03.085","volume":"217","author":"R Almeida","year":"2010","unstructured":"Almeida R, Torres DFM (2010) Leitmann\u2019s direct method for fractional optimization problems. Appl Math Comput 217(3):956\u2013962","journal-title":"Appl Math Comput"},{"issue":"3","key":"9_CR8_9","doi-asserted-by":"publisher","first-page":"1490","DOI":"10.1016\/j.cnsns.2010.07.016","volume":"16","author":"R Almeida","year":"2011","unstructured":"Almeida R, Torres DFM (2011) Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Comm Nonlinear Sci Numer Simult 16(3):1490\u20131500","journal-title":"Comm Nonlinear Sci Numer Simult"},{"issue":"9","key":"9_CR9_9","doi-asserted-by":"crossref","first-page":"095201","DOI":"10.1088\/1751-8113\/41\/9\/095201","volume":"41","author":"TM Atanackovi\u0107","year":"2007","unstructured":"Atanackovi\u0107 TM, Konjik S, Pilipovi\u0107 S (2007) Variational problems with fractional derivatives: Euler-Lagrange equations. J Phys A 41(9):095201, 12 pp","journal-title":"J Phys A"},{"issue":"2","key":"9_CR10_9","doi-asserted-by":"publisher","first-page":"021102","DOI":"10.1115\/1.2833586","volume":"3","author":"D Baleanu","year":"2008","unstructured":"Baleanu D (2008) Fractional constrained systems and caputo derivatives. J Comput Nonlinear Dynam 3(2):021102","journal-title":"J Comput Nonlinear Dynam"},{"issue":"2","key":"9_CR11_9","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/S0034-4877(08)80007-9","volume":"61","author":"D Baleanu","year":"2008","unstructured":"Baleanu D (2008) New applications of fractional variational principles. Rep Math Phys 61(2):199\u2013206","journal-title":"Rep Math Phys"},{"key":"9_CR12_9","doi-asserted-by":"publisher","DOI":"10.1007\/978-90-481-3293-5","volume-title":"New trends in nanotechnology and fractional calculus applications","author":"D Baleanu","year":"2010","unstructured":"Baleanu D, G\u00fcven\u00e7 ZB, Tenreiro Machado JA (2010) New trends in nanotechnology and fractional calculus applications. Springer, New York"},{"issue":"2","key":"9_CR13_9","doi-asserted-by":"publisher","first-page":"417","DOI":"10.3934\/dcds.2011.29.417","volume":"29","author":"NRO Bastos","year":"2011","unstructured":"Bastos NRO, Ferreira RAC, Torres DFM (2011) Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin Dyn Syst 29(2):417\u2013437","journal-title":"Discrete Contin Dyn Syst"},{"issue":"3","key":"9_CR14_9","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1016\/j.sigpro.2010.05.001","volume":"91","author":"NRO Bastos","year":"2011","unstructured":"Bastos NRO, Ferreira RAC, Torres DFM (2011) Discrete-time fractional variational problems. Signal Process 91(3):513\u2013524","journal-title":"Signal Process"},{"issue":"15","key":"9_CR15_9","doi-asserted-by":"publisher","first-page":"1931","DOI":"10.1002\/mma.879","volume":"30","author":"RA El-Nabulsi","year":"2007","unstructured":"El-Nabulsi RA, Torres DFM (2007) Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (\u03b1,\u2009\u03b2). Math Methods Appl Sci 30(15):1931\u20131939","journal-title":"Math Methods Appl Sci"},{"issue":"5","key":"9_CR16_9","doi-asserted-by":"crossref","first-page":"053521","DOI":"10.1063\/1.2929662","volume":"49","author":"RA El-Nabulsi","year":"2008","unstructured":"El-Nabulsi RA, Torres DFM (2008) Fractional actionlike variational problems. J Math Phys 49(5):053521, 7 pp","journal-title":"J Math Phys"},{"issue":"2","key":"9_CR17_9","doi-asserted-by":"publisher","first-page":"834","DOI":"10.1016\/j.jmaa.2007.01.013","volume":"334","author":"GSF Frederico","year":"2007","unstructured":"Frederico GSF, Torres DFM (2007) A formulation of Noether\u2019s theorem for fractional problems of the calculus of variations. J Math Anal Appl 334(2):834\u2013846","journal-title":"J Math Anal Appl"},{"issue":"3","key":"9_CR18_9","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1007\/s11071-007-9309-z","volume":"53","author":"GSF Frederico","year":"2008","unstructured":"Frederico GSF, Torres DFM (2008) Fractional conservation laws in optimal control theory. Nonlinear Dynam 53(3):215\u2013222","journal-title":"Nonlinear Dynam"},{"issue":"3","key":"9_CR19_9","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1016\/j.amc.2010.01.100","volume":"217","author":"GSF Frederico","year":"2010","unstructured":"Frederico GSF, Torres DFM (2010) Fractional Noether\u2019s theorem in the Riesz-Caputo sense. Appl Math Comput 217(3):1023\u20131033","journal-title":"Appl Math Comput"},{"key":"9_CR20_9","volume-title":"Theory and applications of fractional differential equations","author":"AA Kilbas","year":"2006","unstructured":"Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam"},{"key":"9_CR21_9","unstructured":"Klimek M (2009) On solutions of linear fractional differential equations of a variational type. Czestochowa Series Monographs 172, Czestochowa University of Technology, Czestochowa"},{"issue":"3","key":"9_CR22_9","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1016\/j.sigpro.2010.08.003","volume":"91","author":"R Magin","year":"2011","unstructured":"Magin R, Ortigueira MD, Podlubny I, Trujillo J (2011) On the fractional signals and systems. Signal Process 91(3):350\u2013371","journal-title":"Signal Process"},{"issue":"14","key":"9_CR23_9","doi-asserted-by":"publisher","first-page":"1712","DOI":"10.1002\/mma.1289","volume":"33","author":"AB Malinowska","year":"2010","unstructured":"Malinowska AB, Torres DFM (2010) Natural boundary conditions in the calculus of variations. Math Methods Appl Sci 33(14):1712\u20131722","journal-title":"Math Methods Appl Sci"},{"issue":"9","key":"9_CR24_9","doi-asserted-by":"publisher","first-page":"3110","DOI":"10.1016\/j.camwa.2010.02.032","volume":"59","author":"AB Malinowska","year":"2010","unstructured":"Malinowska AB, Torres DFM (2010) Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput Math Appl 59(9):3110\u20133116","journal-title":"Comput Math Appl"},{"issue":"3","key":"9_CR25_9","doi-asserted-by":"publisher","first-page":"1158","DOI":"10.1016\/j.amc.2010.01.015","volume":"217","author":"AB Malinowska","year":"2010","unstructured":"Malinowska AB, Torres DFM (2010) Leitmann\u2019s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales. Appl Math Comput 217(3):1158\u20131162","journal-title":"Appl Math Comput"},{"issue":"3","key":"9_CR26_9","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1007\/s10957-010-9730-1","volume":"147","author":"AB Malinowska","year":"2010","unstructured":"Malinowska AB, Torres DFM (2010) The Hahn quantum variational calculus. J Optim Theory Appl 147(3):419\u2013442","journal-title":"J Optim Theory Appl"},{"issue":"2","key":"9_CR27_9","first-page":"210","volume":"26","author":"D Mozyrska","year":"2010","unstructured":"Mozyrska D, Torres DFM (2010) Minimal modified energy control for fractional linear control systems with the Caputo derivative. Carpathian J Math 26(2):210\u2013221","journal-title":"Carpathian J Math"},{"issue":"3","key":"9_CR28_9","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1016\/j.sigpro.2010.07.016","volume":"91","author":"D Mozyrska","year":"2011","unstructured":"Mozyrska D, Torres DFM (2011) Modified optimal energy and initial memory of fractional continuous-time linear systems. Signal Process 91(3):379\u2013385","journal-title":"Signal Process"},{"key":"9_CR29_9","doi-asserted-by":"crossref","unstructured":"Odzijewicz T, Malinowska AB, Torres DFM (2011) Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal (in press) DOI: 10.1016\/j.na.2011.01.010","DOI":"10.1016\/j.na.2011.01.010"},{"key":"9_CR30_9","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations. Academic Press, San Diego, CA"},{"issue":"3","key":"9_CR31_9","doi-asserted-by":"publisher","first-page":"3581","DOI":"10.1103\/PhysRevE.55.3581","volume":"55","author":"F Riewe","year":"1997","unstructured":"Riewe F (1997) Mechanics with fractional derivatives. Phys Rev E 55(3):3581\u20133592","journal-title":"Phys Rev E"},{"issue":"1","key":"9_CR32_9","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s10957-007-9292-z","volume":"137","author":"DFM Torres","year":"2008","unstructured":"Torres DFM, Leitmann G (2008) Contrasting two transformation-based methods for obtaining absolute extrema. J Optim Theory Appl 137(1):53\u201359","journal-title":"J Optim Theory Appl"},{"key":"9_CR33_9","doi-asserted-by":"crossref","DOI":"10.1007\/b97436","volume-title":"The calculus of variations","author":"B van Brunt","year":"2004","unstructured":"van Brunt B (2004) The calculus of variations. Universitext, Springer, New York"}],"container-title":["Fractional Dynamics and Control"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-1-4614-0457-6_9","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,18]],"date-time":"2023-02-18T20:50:17Z","timestamp":1676753417000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-1-4614-0457-6_9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,21]]},"ISBN":["9781461404569","9781461404576"],"references-count":33,"URL":"https:\/\/doi.org\/10.1007\/978-1-4614-0457-6_9","relation":{},"subject":[],"published":{"date-parts":[[2011,10,21]]},"assertion":[{"value":"21 October 2011","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}