Abstract
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on the derivative-order α.
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O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, No 1 (2002), 368–379.; DOI: 10.1016/S0022-247X(02)00180-4.
H.M. Ali, F. Lobo Pereira, S.M.A. Gama, A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems. Math. Methods Appl. Sci. 39, No 13 (2016), 3640–3649.; DOI: 10.1002/mma.3811.
R. Almeida, R.A.C. Ferreira, D.F.M. Torres, Isoperimetric problems of the calculus of variations with fractional derivatives. Acta Math. Sci. Ser. B (Engl. Ed.) 32, No 2 (2012), 619–630; DOI: 10.1016/S0252-9602(12)60043-5.
T.M. Atanackovic, S. Konjik, S. Pilipovic, Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A 41, No 9 (2008), Art. 095201, 12pp; DOI: 10.1088/1751-8113/41/9/095201.
L. Bourdin, D. Idczak, A fractional fundamental lemma and a fractional integration by parts formula—Applications to critical points of Bolza functionals and to linear boundary value problems. Adv. Differential Equations 20, No 3-4 (2015), 213–232; https://projecteuclid.org/euclid.ade/1423055200.
J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48, No 3 (2007), Art. 033504, 34 pp; DOI: 10.1063/1.2483292.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, (2006).
M.J. Lazo, D.F.M. Torres, The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of Caputo. J. Optim. Theory Appl. 156, No 1 (2013), 56–67.; DOI: 10.1007/s10957-012-0203-6.
A. Malinowska, D.F.M. Torres, Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 14, No 4 (2011), 523–537.; DOI: 10.2478/s13540-011-0032-6; https://www.degruyter.com/view/j/fca.2011.14.issue-4/issue-files/fca.2011.14.issue-4.xml.
J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, 74, Springer-Verlag, New York, (1989).
F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 3 53, No 2 (1996), 1890–1899; DOI: 10.1103/PhysRevE.53.1890.
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives (Transl. from the 1987 Russian original) Gordon and Breach Science Publishers, Yverdon, (1993).
B. van Brunt, The Calculus of Variations. Universitext, Springer-Verlag, New York, (2004).
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Ferreira, R.A.C. Fractional Calculus of Variations: A Novel Way to Look At It. FCAA 22, 1133–1144 (2019). https://doi.org/10.1515/fca-2019-0059
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DOI: https://doi.org/10.1515/fca-2019-0059