Noether-type theorem for fractional variational problems depending on fractional derivatives of functions

References

• Oldham KB, Spanier J, editors. The fractional calculus. New York: Academic Press; 1974.
   Google Scholar
• Sabatier J, Agrawal OP, Tenreiro Machado JA. Advances in fractional calculus: theoretical developments and applications in physics and engineering. Dordrecht, Netherlands: Springer; 2007.
   Google Scholar
• Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier; 2006.
   Google Scholar
• Diethelm K. The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Berlin: Springer-Verlag; 2010.
   Google Scholar
• Hilfer R, editor. Applications of fractional calculus in physics. Singapore: World Scientific; 2000.
   Google Scholar
• Magin RL. Fractional calculus in bioengineering. United States: Begell House; 2006. https://doi.org/10.1109/CarpathianCC.2012.6228688.
   Google Scholar
• Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives – theory and applications. Linghorne (PA): Gordon and Breach; 1993.
   Google Scholar
• Riewe F. Nonconservative Lagrangian and Hamiltonian mechanics. Phys Rev E. 1996;53:1890–1899. Available from: https://doi.org/10.1103/PhysRevE.53.1890
   PubMed Web of Science ®Google Scholar
• Riewe F. Mechanics with fractional derivatives. Phys Rev E. 1997;55:3581–3592. Available from: https://doi.org/10.1103/PhysRevE.55.3581
   Web of Science ®Google Scholar
• Bauer PS. Dissipative dynamical systems I. Proc Natl Acad Sci. 1931;17:311–314. Available from: https://doi.org/10.1073/pnas.17.5.311
   PubMedGoogle Scholar
• Lazo MJ, Krumreich CE. The action principle for dissipative systems. J Math Phys. 2014;55:122902. Available from: https://doi.org/10.1063/1.4903991
   Web of Science ®Google Scholar
• Malinowska AB, Torres DFM. Introduction to the fractional calculus of variations. London: Imperial College Press; 2012.
   Google Scholar
• Frederico GSF, Lazo MJ. Fractional Noether's theorem with classical and Caputo derivatives: constants of motion for non-conservative system. Nonlinear Dyn. 2016;85:839–851. Available from: https://doi.org/10.1007/s11071-016-2727-z
   Web of Science ®Google Scholar
• El-Nabulsi RA. Fractional variational symmetries of Lagrangians, the fractional Galilean transformation and the modified Schrödinger equation. Nonlinear Dyn. 2015;81:939–948. Available from: https://doi.org/10.1007/s11071-015-2042-0
   Web of Science ®Google Scholar
• Frederico GSF, Torres DFM. Conservation laws for invariant functionals containing compositions. Appl Anal. 2007;86(9):1117–1126. Available from: https://doi.org/10.1080/00036810701584583
   Web of Science ®Google Scholar
• Frederico GSF, Torres DFM. Non-conservative Noether's theorem for fractional action-like variational problems with intrinsic and observer times. Int J Ecol Econ Stat. 2007;9(F07):74–82. Available from: http://www.ceser.in/ceserp/index.php/ijees/article/view/1818
   Google Scholar
• Frederico GSF, Torres DFM. A formulation of Noether's theorem for fractional problems of the calculus of variations. J Math Anal Appl. 2007;334(2):834–846. Available from: https://doi.org/10.1016/j.jmaa.2007.01.013
   Web of Science ®Google Scholar
• Frederico GSF, Torres DFM. Fractional optimal control in the sense of Caputo and the fractional Noether's theorem. Int Math Forum. 2008;3(10):479–493. arXiv:0712.1844
   Google Scholar
• Frederico GSF, Torres DFM. Fractional conservation laws in optimal control theory. Nonlinear Dyn. 2008;53(3):215–222. Available from: https://doi.org/10.1007/s11071-007-9309-z
   Web of Science ®Google Scholar
• Zhang Y, Zhai XH. Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dyn. 2015;81:469–480. Available from: https://doi.org/10.1007/s11071-015-2005-5
   Web of Science ®Google Scholar
• Lazo MJ. Fractional variational problems depending on fractional derivatives of differentiable functions with application to nonlinear chaotic systems. Conf Pap Math. 2013;2013:872869. Available from: http://dx.doi.org/10.1155/2013/872869
   Google Scholar
• Sousa JVC, de Oliveira EC. On the ψ-Hilfer fractional derivative. Commun Nonlinear Sci. 2018;60:72–91. Available from: https://doi.org/10.1016/j.cnsns.2018.01.005
   Web of Science ®Google Scholar
• Sousa JVC, de Oliveira EC. Leibniz type rule: ψ-Hilfer fractional operator. Commun Nonlinear Sci. 2019;77:305–311. Available from: https://doi.org/10.1016/j.cnsns.2019.05.003
   Web of Science ®Google Scholar
• Schot SH. Jerk: the time rate of change of acceleration. Am J Phys. 1978;46:1090–1094. Available from: https://doi.org/10.1119/1.11504
   Web of Science ®Google Scholar
• Sprott JC. Elegant chaos. Singapore: World Scientific; 2010.
   Google Scholar
• Gottlieb HPW. What is the simplest jerk function that gives chaos? Am J Phys. 1996;64:525–525. Available from: https://doi.org/10.1119/1.18276
   Web of Science ®Google Scholar
• Sprott JC. Some simple chaotic jerk functions. Am J Phys. 1997;65:537–543. Available from: https://doi.org/10.1119/1.18585
   Web of Science ®Google Scholar
• Linz SJ. Nonlinear dynamical models and jerky motion. Am J Phys. 1997;65:523–526. Available from: https://doi.org/10.1119/1.18594
   Web of Science ®Google Scholar
• Sprott JC. Simplest dissipative chaotic flow. Phys Lett A. 1997;228:271–274. Available from: https://doi.org/10.1016/S0375-9601(97)00088-1
   Web of Science ®Google Scholar
• Linz SJ, Sprott JC. Elementary chaotic flow. Phys Lett A. 1999;259:240–245. Available from: https://doi.org/10.1016/S0375-9601(99)00450-8
   Web of Science ®Google Scholar
• Sprott JC. A new class of chaotic circuit. Phys Lett A. 2000;266:19–23. Available from: https://doi.org/10.1016/S0375-9601(00)00026-8
   Web of Science ®Google Scholar
• Patidar V, Sud KK. Identical synchronization in chaotic jerk dynamical systems. Electron J Theor Phys. 2006;3(11):33–70. Available from: http://www.ejtp.com/articles/ejtpv3i11p33.pdf
   Google Scholar
• Teodoro GS, Machado JAT, de Oliveira EC. A review of definitions of fractional derivatives and other operators. J Comput Phys. 2019;388:195–208. Available from: https://doi.org/10.1016/j.jcp.2019.03.008
   Web of Science ®Google Scholar
• Agrawal OP. Formulation of Euler-Lagrange equations for fractional variational problems. J Math Anal Appl. 2002;272:368–379. Available from: https://doi.org/10.1016/S0022-247X(02)00180-4
   Web of Science ®Google Scholar
• Avez A. Differential calculus. New York City, United States: John Wiley Sons; 1986.
   Google Scholar
• Bourdin L. A class of fractional optimal control problems and fractional Pontryagin's systems. Existence of a fractional Noether's theorem. preprint, 2012. arXiv:1203.1422.
   Google Scholar