Herglotz Type Variational Problems on Conformable Calculus

Serkan Aslıyüce; Süleyman Öğrekçi

doi:10.47000/tjmcs.1641418

Herglotz Type Variational Problems on Conformable Calculus

Abstract

This paper presents generalized Euler-Lagrange equations involving fractional order derivatives for Herglotz type problems. Unlike traditional functionals, the extremals of these problems are derived from differential equations. The fractional generalized Euler-Lagrange equation obtained in this paper includes both the classical generalized Euler-Lagrange equation and the Euler-Lagrange equation obtained for variational problems involving conformable fractional derivatives as special cases. We use this equation to prove the Noether-type theorem for invariant functionals.

Keywords

Ethical Statement

The author declares that there are no conflicts of interest regarding the publication of this article.

References

Abdelhakim, A.A., The flaw in the conformable calculus: it is conformable because it is not fractional, Fract. Calc. Appl. Anal., 22(2)(2019), 242–254.
Abdelhakim, A.A., Machadoi J.A.T., A critical analysis of the conformable derivative, Nonlinear Dynam., 95(2019), 3063–3073.
Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279(2015), 57–66.
Agarwal, O.P., Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272(2002), 368–379.
Agarwal, O.P., Fractional variational calculus and the transversality conditions, J. Phys. A, 39(33)(2006), 10375–10384.
Agarwal, O.P., Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40(24)(2007), 6287–6303.
Almeida, R., Fractional variational problems with the Riesz-Caputo derivative, Appl. Math. Lett., 25(2)(2012), 142–148.
Almeida, R., Variational problems involving a Caputo-type fractional derivative, J. Optim. Theory Appl., 174(1)(2017), 276–294.

Bastos, N.R.O., Calculus of variations involving Caputo-Fabrizio fractional differentiation, Stat., Optim. Inf. Comput., 6(2018), 12–21.
Chatibi, Y., El Kinani, E.H., Ouhadan, A., Variational calculus involving nonlocal fractional derivative with Mittag-Leffer kernel, Chaos, Solitons & Fractals, 118(2019), 117–121.
Eroğlu, B.B.I., Yapışkan, D., Local generalization of transversality conditions for optimal control problem, Math. Model. Nat. Phenom., 14(3)(2019), 310.
Garra, R., Taverna, G.S., Torres, D.F.M., Fractional Herglotz variational principles with generalized Caputo derivatives, Chaos, Solitons & Fractals, 102(2017), 94–98.
Gelfand, I.M., Fomin, S.V., Calculus of Variations, Prentice-Hall Inc., 1963.
Georgieva, B., Guenther, R.B., First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20(2)(2002), 261–273.
Georgieva, B., Guenther, R., Bodurov, T., Generalized variational principle of Herglotz for several independent variables, First Noether-type theorem, Journal of Mathematical Physics, 44(9)(2003), 3911–3927.
Georgieva, B., Guenther, R.B., Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26(2)(2005), 307–314.
Guenther, R.B., Gottsch, J.A., Kramer, D.B., The Herglotz algorithm for constructing canonical transformations, SIAM Review, 38(2)(1996), 287–293.
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. Nicholas Copernicus University, Torun, 1996.
Herglotz, G., Beruhrungstransformationen, Lectures at the University of Gottingen, 1930.
İskender Eroğlu, B.B., Yapışkan, D., Generalized conformable variational calculus and optimal control problems with variable terminal conditions, AIMS Mathematics, 5(2020), 1105–1126.
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264(2014), 65–70.
Lazo, M.J., Torres, D.F.M., Variational calculus with conformable fractional derivatives, IEEE/CAA J. Autom. Sin., 4(2)(2017), 340–352.
Ortigueira, M.D., Machado, J.A.T., What is a fractional derivative?, J. Comput. Phys., 293(2015), 4–13.
Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53(2)(1996), 1890–1899.
Riewe, F., Mechanics with fractional derivatives, Phys. Rev. E, 55(3)(1997), 3581–3592.
Santos, S.P.S., Martins, N., Torres, D.F.M., Higher-order variational problems of Herglotz type, Vietnam Journal of Mathematics, 42(4)(2014), 409–419.
Santos, S.P.S., Martins, N., Torres, D.F.M., An optimal control approach to Herglotz variational problems, In Alexander Plakhov, Tatiana Tchemisova, and Adelaide Freitas, editors, Optimization in the Natural Sciences, pages 107-117, Cham, Springer International Publishing, 2015.
Stynes, M., Fractional-order derivatives defined by continuous kernels are too restrictive, Appl. Math. Lett., 85(2018), 22–26.
Tarasov, V.E., No nonlocality. No fractional derivative, Commun Nonlinear Sci. Numer. Simul., 62(2018), 157–163.
Tarasov, V.E., “Conformable fractional” derivatives and integrals are integer-order operators: Physical and geometrical interpretations, applications to fractal physics, Chaos, Solitons & Fractals, 192(2025), 116066.
Yazıcı, D., Topuz, S., Generalized fractional bi-Hamiltonian structure of Pleba´nski’s second heavenly equation in terms of conformable fractional derivatives, J. Comput. Appl. Math, 452(2024), 116121.
Zhang, J., Ma, X., Li, L., Optimality conditions for fractional variational problems with Caputo-Fabrizio fractional derivatives, Adv. Differ. Equ., (2017), 357.

Details

Primary Language

English

Subjects

Calculus of Variations, Mathematical Aspects of Systems Theory and Control Theory

Journal Section

Research Article

Authors

Serkan Aslıyüce ^*
0000-0003-1729-3914
Türkiye

Süleyman Öğrekçi
0000-0003-1205-6848
Türkiye

Publication Date

June 30, 2026

Submission Date

February 17, 2025

Acceptance Date

December 24, 2025

Published in Issue

Year 2026 Volume: 18 Number: 2

DOI

https://doi.org/10.47000/tjmcs.1641418

IZ

https://izlik.org/JA84TM43LP

Cite

RIS / Bibtex

APA

Aslıyüce, S., & Öğrekçi, S. (2026). Herglotz Type Variational Problems on Conformable Calculus. Turkish Journal of Mathematics and Computer Science, 18(2), 310-318. https://doi.org/10.47000/tjmcs.1641418

AMA

1.Aslıyüce S, Öğrekçi S. Herglotz Type Variational Problems on Conformable Calculus. TJMCS. 2026;18(2):310-318. doi:10.47000/tjmcs.1641418

Chicago

Aslıyüce, Serkan, and Süleyman Öğrekçi. 2026. “Herglotz Type Variational Problems on Conformable Calculus”. Turkish Journal of Mathematics and Computer Science 18 (2): 310-18. https://doi.org/10.47000/tjmcs.1641418.

EndNote

Aslıyüce S, Öğrekçi S (June 1, 2026) Herglotz Type Variational Problems on Conformable Calculus. Turkish Journal of Mathematics and Computer Science 18 2 310–318.

IEEE

[1]S. Aslıyüce and S. Öğrekçi, “Herglotz Type Variational Problems on Conformable Calculus”, TJMCS, vol. 18, no. 2, pp. 310–318, June 2026, doi: 10.47000/tjmcs.1641418.

ISNAD

Aslıyüce, Serkan - Öğrekçi, Süleyman. “Herglotz Type Variational Problems on Conformable Calculus”. Turkish Journal of Mathematics and Computer Science 18/2 (June 1, 2026): 310-318. https://doi.org/10.47000/tjmcs.1641418.

JAMA

1.Aslıyüce S, Öğrekçi S. Herglotz Type Variational Problems on Conformable Calculus. TJMCS. 2026;18:310–318.

MLA

Aslıyüce, Serkan, and Süleyman Öğrekçi. “Herglotz Type Variational Problems on Conformable Calculus”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 2, June 2026, pp. 310-8, doi:10.47000/tjmcs.1641418.

Vancouver

1.Serkan Aslıyüce, Süleyman Öğrekçi. Herglotz Type Variational Problems on Conformable Calculus. TJMCS. 2026 Jun. 1;18(2):310-8. doi:10.47000/tjmcs.1641418