Advanced Search
Article Contents
Article Contents

Fractional variational principle of Herglotz

Abstract Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


    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.


    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.


    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.


    B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.


    B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.


    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.


    H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, MA, 1951.


    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.


    G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930.


    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.


    M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of Czenstochowa University of Technology, Czestochowa, 2009.


    C. Lánczos, The Variational Principles of Mechanics, Fourth edition, Mathematical Expositions, No. 4, Univ. Toronto Press, Toronto, ON, 1970.


    A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imp. Coll. Press, London, 2012.doi: 10.1142/p871.


    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.


    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.


    I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.


    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.


    F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899.doi: 10.1103/PhysRevE.53.1890.


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(127) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint