Article Contents
Article Contents

# Fractional variational principle of Herglotz

• The aim of this paper is to bring together two approaches to non-conservative systems -- the generalized variational principle of Herglotz and the fractional calculus of variations. Namely, we consider functionals whose extrema are sought, by differential equations that involve Caputo fractional derivatives. The Euler--Lagrange equations are obtained for the fractional variational problems of Herglotz-type and the transversality conditions are derived. The fractional Noether-type theorem for conservative and non-conservative physical systems is proved.
Mathematics Subject Classification: Primary: 49K05; Secondary: 26A33.

 Citation:

•  [1] D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines and Y. S. Ilyashenko, Ordinary Differential Equations and Smooth Dynamical Systems, Encyclopaedia of Mathematical Sciences, 1, Springer, Berlin, 1988. [2] T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations, J. Phys. A, 41 (2008), 095201, 12 pp.doi: 10.1088/1751-8113/41/9/095201. [3] T. M. Atanacković and B. Stanković, On a numerical scheme for solving differential equations of fractional order, Mech. Res. Comm., 35 (2008), 429-438.doi: 10.1016/j.mechrescom.2008.05.003. [4] G. S. F. Frederico and D. F. M. Torres, A formulation of Noethers theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., 334 (2007), 834-846.doi: 10.1016/j.jmaa.2007.01.013. [5] N. J. Ford and M. L. Morgado, Fractional boundary value problems: Analysis and numerical methods, Fract. Calc. Appl. Anal., 14 (2011), 554-567.doi: 10.2478/s13540-011-0034-4. [6] N. J. Ford and M. L. Morgado, Distributed order equations as boundary value problems, Comput. Math. Appl., 64 (2012), 2973-2981.doi: 10.1016/j.camwa.2012.01.053. [7] B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273. [8] B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314. [9] B. Georgieva, R. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927.doi: 10.1063/1.1597419. [10] H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, MA, 1951. [11] R. B. Guenther, C. M. Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Trans- formations and Hamiltonian Systems, Lecture Notes in Nonlinear Analysis, 1, Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Toruń, 1996. [12] G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930. [13] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006. [14] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of Czenstochowa University of Technology, Czestochowa, 2009. [15] C. Lánczos, The Variational Principles of Mechanics, Fourth edition, Mathematical Expositions, No. 4, Univ. Toronto Press, Toronto, ON, 1970. [16] A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imp. Coll. Press, London, 2012.doi: 10.1142/p871. [17] A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Appl. Math. Lett., 25 (2012), 1941-1946.doi: 10.1016/j.aml.2012.03.006. [18] V. J. Menon, N. Chanana and Y. Singh, A fresh look at the BCK frictional lagrangian, Prog. Theor. Phys., 98 (1997), 321-329.doi: 10.1143/PTP.98.321. [19] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. [20] S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives, Comput. Math. Appl., 64 (2012), 3090-3100.doi: 10.1016/j.camwa.2012.01.068. [21] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899.doi: 10.1103/PhysRevE.53.1890. [22] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993. [23] S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type, Vietnam Journal of Mathematics, (2013), 1-11.doi: 10.1007/s10013-013-0048-9.