THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

Jing Li Fu

,

Lijun Zhang

,

Chaudry Masood Khalique

,

Ma Li Guo

online first only

Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative

ABSTRACT
In this paper, we present the fractional motion equations and fractional non-Noether symmetries of Lagrangian systems with the conformable fractional derivatives. The exchanging relationship between isochronous variation and fractional derivative, and the fractional Hamilton's principle of the holonomic conservative and non-conservative systems under the conformable fractional derivative are proposed. Then the fractional motion equations of these systems based on the Hamilton's principle are established. The fractional Euler operator, the definition of fractional non-Noether symmetries, non-Noether theorem and Hojman's conserved quantities for the Lagrangian systems are obtained with conformable fractional derivative. An example is given to illustrate the results.
KEYWORDS
non-Noether symmetry, Lagrangian system, Conformable fractional derivative
PAPER SUBMITTED: 2020-05-20
PAPER REVISED: 2020-06-20
PAPER ACCEPTED: 2020-06-20
PUBLISHED ONLINE: 2021-01-31
DOI REFERENCE: https://doi.org/10.2298/TSCI200520035F
REFERENCES
  1. G. W. Bluman, A.F. Cheviakov, S. Anco, Applications of symmetry methods to partial differential equations. New York: Springer, 2009, 10-20
  2. R. Dastranj, M. Nadjafikhah, Symmetry analysis and conservation laws for description of waves in bubbly liquid. Int. J. Non-Linear Mech. 2014, 67: 48-51
  3. S. A. Hojman, A new conservation law constructed without using either Lagrangians or Hamiltonians. J. Phys. A: Math. Gen., 1992, 25: 291-295
  4. F. Gonzalez-Gascon, Geometric foundations of a new conservation law discovered by Hojman. J. Phys. A: Math. Gen., 1994, 27: 59-60
  5. L. Zhang, P. Yuan, J. Fu, C M Khalique. Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation. Disc. Cont. Dyna. Syst. - S, doi: 10.3934/dcdss.2020214
  6. J. Cai, S.Luo, F. Mei, Conformal invariance and conserved quantity of Hamilton systems. Chin. Phya. B, 2008, 17: 3170-3174
  7. J. Fu, L. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems. Phys. Lett.A, 2003, 317: 255-259
  8. J. Fu , H. Fu, R. Liu, Hojman conserved quantities of discrete mechanico-electrical systems constructed by continuous symmetries. Phys. Lett. A, 2010, 374 (2010): 1812-1818
  9. J. Fu, L.Chen, Non-Noether symmetries and conserved quantities of Lagrange mechanicoelectrical systems, Chin. Phys., 2004, 13: 1784-1789
  10. S. Zhou, H. Fu, J. Fu, Symmetry theories of Hamiltonian systems with fractional derivative. Sci. Chin. Phys., Mech. Astron., 2011, 54(10): 1847--1853
  11. S. Zhang, B. Chen, J. Fu, Hamilton formalism and Noether symmetry for mechanico--electrical systems with fractional derivatives. Chin. Phys. B, 2012, 21: 100202
  12. Y. Sun, B. Chen, J. Fu. Lie symmetry theorem of fractional nonholonomic systems, Chin. Phys. B, 2014, 23(11): 110201
  13. L. Wang, J. Fu. Fractional cyclic integrals and Routh equations of fractional Lagrange system with combined Caputo derivatives, Chin. Phys. B,2014, 23 (12): 124501
  14. J. Fu, L. Fu, B. Chen, Y. Sun. Lie symmetries and their inverse problems of nonholonomic Hamilton systems with fractional derivatives. Physics Letters A 2016, 380(1--2): 15--21
  15. Y. Zhou, Y. Zhang, Noether's theorems of a fractional Birkhoffian system within Riemann Liouville derivatives. Chin. Phys. B, 2014, 23: 124502
  16. I. Podlubny, Fractional Differential Equations, Academic, San Diego: Academic Press, 1999
  17. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific River Edge, USA, 2000
  18. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier B V, 2006
  19. O. P. Agrawal, Formulation of Euler--Lagranian equations for fractional variational problems. J. Math. Anal. Appl., 2002, 272: 368--379
  20. Y. Zhang, Fractional differential equations of motion in terms of combined Riemann--Liouville derivatives, Chin. Phys. B, 2012, 21: 084502
  21. T. Abdeljawad. On conformable fractional calculus, J. Comput. Appl. Math., 2015, 279: 57--66
  22. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math., 2014, 264: 65-70
PDF VERSION [DOWNLOAD]
Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative