Variational problems of Herglotz type with complex order fractional derivatives and less regular Lagrangian

  • Teodor M. Atanacković
  • Sanja KonjikEmail author
  • Stevan Pilipović
Original Paper


We derive optimality conditions for variational problems of Herglotz type whose Lagrangian depends on fractional derivatives of both real and complex order, and resolve the case of subdomain when the lower bounds of variational integral and fractional derivatives differ. Moreover, we consider a problem of the Herglotz type that corresponds to the case when the Lagrangian depends on the fractional derivative of the action and give an example of the problem that corresponds to the oscillator with a memory. Since our assumptions on the Lagrangian are weaker than in the classical theory, we analyze generalized Euler–Lagrange equations by the use of weak derivatives and the appropriate technics of distribution theory. Such an example is discussed in detail.

Mathematics Subject Classification

Primary 26A33 Secondary 49K05 34A08 



This work is supported by Projects 174005 and 174024 of the Serbian Ministry of Education, Science, and Technological Development and Project 142-451-2384 of the Provincial Secretariat for Higher Education and Scientific Research.


  1. 1.
    Almeida, R., Malinowska, A.B.: Fractional variational principle of Herglotz. Discrete Contin. Dyn. Syst. Ser. B 19(8), 2367–2381 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atanacković, T.M., Janev, M., Pilipović, S., Zorica, D.: Euler–Lagrange equations for Lagrangians containing complex order fractional derivatives. J. Optim. Theory Appl. 174(1), 256–275 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atanacković, T.M., Konjik, S., Pilipović, S.: Variational problems with fractional derivatives: Euler–Lagrange equations. J. Phys. A Math. Theor. 41(9), 095201–095213 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Atanacković, T.M., Konjik, S., Pilipović, S., Zorica, D.: Complex order fractional derivatives in viscoelasticity. Mech. Time Depend. Mater. 20(2), 175–195 (2016)CrossRefGoogle Scholar
  5. 5.
    Bourdin, L., Idczak, D.: 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. Differ. Equ. 20(3–4), 213–232 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cislo, J., Lopuszariski, J.T., Stichel, P.C.: On the inverse variational problem in classical mechanics. In: Rernbielinski, J. (ed.) Particles, Fields and Gravitation. AIP Conference Proceedings/High Energy Physics, vol. 453, pp. 219–225. American Institute of Physics (1998)Google Scholar
  7. 7.
    Garra, R., Taverna, G.S., Torres, D.F.M.: Fractional Herglotz variational principles with generalized Caputo derivatives. Chaos Solitons Fractals 102, 94–98 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  9. 9.
    Guenther, R.B., 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, Toruń (1995)Google Scholar
  10. 10.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  11. 11.
    Love, E.R.: Fractional derivatives of imaginary order. J. Lond. Math. Soc. 2–3(2), 241–259 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993)zbMATHGoogle Scholar
  13. 13.
    Tavares, D., Almeida, R., Torres, D.F.M.: Fractional Herglotz variational problems of variable order. Discrete Contin. Dyn. Syst. Ser. S 11, 143–154 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Tian, X., Zhang, Y.: Noether symmetry and conserved quantity for Hamiltonian system of Herglotz type on time scales. Acta Mech. 229(9), 3601–3611 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Vujanović, B.D., Atanacković, T.M.: An Introduction to Modern Variational Techniques in Mechanics and Engineering. Birkhäuser, Boston (2004)CrossRefzbMATHGoogle Scholar
  16. 16.
    Yang, X.-J.: General Fractional Derivatives: Theory, Methods and Applications. CRC Press, New York (2019)CrossRefzbMATHGoogle Scholar
  17. 17.
    Zhang, Y.: Variational problem of Herglotz type for Birkhoffian system and its Noether’s theorems. Acta Mech. 228(4), 1481–1492 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Technical SciencesUniversity of Novi SadNovi SadSerbia
  2. 2.Faculty of SciencesUniversity of Novi SadNovi SadSerbia

Personalised recommendations