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Noether symmetries and conserved quantities for fractional Birkhoffian systems

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Abstract

This paper presents the variational problems for fractional Birkhoffian systems, and its corresponding symmetries and conserved quantities are further studied. First, the fractional Pfaff variational problems in terms of Riemann–Liouville fractional derivatives are proposed, and the fractional Pfaff–Birkhoff–d’Alembert principle is established. Thus, the fractional Birkhoff’s equations are derived. Second, we derive two basic formulae for the variation of Pfaff action and give the definitions and criteria of fractional Noether symmetries and quasi-symmetries. Third, we establish the Noether theorems of fractional Birkhoffian systems which reveal the relationship between fractional symmetries and fractional conserved quantities. Some special cases are discussed, showing the universality of our results. Two examples are analyzed to illustrate the applications of the results in detail.

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References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. Miller, K.S., Ross, B.: An Introduction to the Fractional Integrals and Derivatives—Theory and Applications. Wiley, New York (1993)

    Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  4. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  5. Tarasov, V.E.: Fractional Dynamics. Higher Education Press, Beijing (2010)

    Book  MATH  Google Scholar 

  6. Yang, X.J., Baleanu, D., Machado J.A. T.: Systems of Navier–Stokes equations on Cantor sets. Math. Probl. Eng., Article ID 769724 (2013)

  7. Yang, X.J., Baleanu, D., Zhong, W.P.: Approximation solution to diffusion equation on Cantor time-space. Proc. Rom. Acad. A 14(2), 127–133 (2013)

    MathSciNet  Google Scholar 

  8. Yang, X.J., Hristov, J., Srivastava, H.M., Ahmad, B.: Modelling fractal waves on shallow water surfaces via local fractional Korteweg–de vries equation. Abstr. Appl. Anal., Article ID 278672 (2014)

  9. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  10. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581–3592 (1997)

    Article  MathSciNet  Google Scholar 

  11. Klimek, M.: Fractional sequential mechanics—models with symmetric fractional derivative. Czechoslov. J. Phys. 51(12), 1348–1354 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czechoslov. J. Phys. 52(11), 1247–1253 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Agrawal, O.P.: A new Lagrangian and a new Lagrange equation of motion for fractionally damped systems. J. Appl. Mech. 68(2), 339–341 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272(1), 368–379 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A Math. Theor. 40, 6287–6303 (2007)

    Article  MATH  Google Scholar 

  16. Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119(1), 73–79 (2003)

    Google Scholar 

  17. Muslih, S.I., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. J. Math. Anal. Appl. 304(2), 599–606 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Baleanu, D., Agrawal, O.P.: Fractional Hamilton formalism within Caputo’s derivative. Czechoslov. J. Phys. 56(10–11), 1087–1092 (2006)

    Article  MathSciNet  Google Scholar 

  19. Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327(2), 891–897 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. El-Nabulsi, A.R.: A fractional approach to nonconservative Lagrangian dynamical systems. Fizika A 14(4), 289–298 (2005)

    Google Scholar 

  21. El-Nabulsi, A.R.: Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann–Liouville derivatives of order (\(\alpha \),\(\beta )\). Math. Methods Appl. Sci. 30(15), 1931–1939 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. El-Nabulsi, A.R., Torres, D.F.M.: Fractional action-like variational problems. J. Math. Phys. 49, 053521 (2008)

    Article  MathSciNet  Google Scholar 

  23. El-Nabulsi, A.R.: Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems. Chaos Solitons Fractals 42(1), 52–61 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. El-Nabulsi, A.R.: Fractional variational problems from extended exponentially fractional integral. Appl. Math. Comput. 217, 9492–9496 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. El-Nabulsi, A.R.: A periodic functional approach to the calculus of variations and the problem of time-dependent damped harmonic oscillators. Appl. Math. Lett. 24(10), 1647–1653 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Herzallah, M.A.E., Muslih, S.I., Baleanu, D., Rabei, E.M.: Hamilton–Jacobi and fractional like action with time scaling. Nonlinear Dyn. 66(4), 549–555 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Muslih, S.I., Baleanu, D.: Formulation of Hamiltonian equations for fractional variational problems. Czechoslov. J. Phys. 55(6), 633–764 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rabei, E.M., Ababneh, B.S.: Hamilton–Jacobi fractional mechanics. J. Math. Anal. Appl. 344(2), 799–805 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Stanislavsky, A.A.: Hamiltonian formalism of fractional systems. Eur. Phys. J. B 49(1), 93–101 (2006)

    Article  MathSciNet  Google Scholar 

  30. Cresson, J.: Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48(3), 033504 (2007)

    Article  MathSciNet  Google Scholar 

  31. Jumarie, G.: Fractional Hamilton–Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost functions. J. Appl. Math. Comput. 23(1–2), 215–228 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. EI-Zalan, H.A., Muslih, S.I., Rabei, E.M., Baleanu, D.: Hamilton formulation for continuous systems with second order derivatives. Int. J. Theor. Phys. 47, 2195–2202 (2008)

    Article  Google Scholar 

  33. Jarad, F., Baleanu, D., Maraaba, A.T.: Hamiltonian formulation of singular Lagrangian on time scales. Chin. Phys. Lett. 25(5), 1720–1723 (2008)

    Article  Google Scholar 

  34. Agrawal, O.P.: Generalized variational problems and Euler–Lagrange equations. Comput. Math. Appl. 59(5), 1852–1864 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. Baleanu, D., Muslih, S.I., Rabei, E.M., Golmankhaneh, Alireza K., Golmankhaneh, Ali K.: On fractional Hamiltonian systems possessing first-class constraints within Caputo derivatives. Rom. Rep. Phys. 63(1), 3–8 (2011)

    Google Scholar 

  36. Torres, D.F.M., Almeida, R.: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1490–1500 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  37. Zhang, Y.: Fractional differential equations of motion in terms of combined Riemann–Liouville derivatives. Chin. Phys. B 21(8), 084502 (2012)

    Article  Google Scholar 

  38. Zhang, Y., Mei, F.X.: Fractional differential equations of motion in terms of Riesz fractional derivatives. Trans. Beijing Int. Technol. 32(7), 766–770 (2012)

    MATH  MathSciNet  Google Scholar 

  39. Zhou, Y., Zhang, Y.: Fractional Pfaff–Birkhoff principle and fractional Birkhoff’s equations in terms of Riemann–Liouville derivatives. Bull. Sci. Technol. 29(3), 4–10 (2013). (in Chinese)

    Google Scholar 

  40. Frederico, G.S.F., Torres, D.F.M.: A formulation of Noether’s theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334(2), 834–846 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  41. Frederico, G.S.F., Torres, D.F.M.: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53(3), 215–222 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  42. Frederico, G.S.F.: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int. Math. Forum 3(10), 479–493 (2008)

    MATH  MathSciNet  Google Scholar 

  43. Frederico, G.S.F., Torres, D.F.M.: Fractional Noether’s theorem in the Riesz–Caputo sense. Appl. Math. Comput. 217(3), 1023–1033 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Atanacković, T.M., Konjik, S., Pilipović, S., Simić, S.: Variational problems with fractional derivatives: invariance conditions and Noether’s theorem. Nonlinear Anal. 71, 1504–1517 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  45. Zhou, S., Fu, H., Fu, J.L.: Symmetry theories of Hamiltonian systems with fractional derivatives. Sci. Chin. Phys. Mech. Astron. 54(10), 1847–1853 (2011)

    Article  Google Scholar 

  46. Frederico, G.S.F., Torres, D.F.M.: Constants of motion for fractional action-like variational problems. Int. J. Appl. Math. 19(1), 97–104 (2006)

    MATH  MathSciNet  Google Scholar 

  47. Frederico, G.S.F., Torres, D.F.M.: Non-conservative Noether’s theorem for fractional action-like variational problems with intrinsic and observer times. Int. J. Ecol. Econ. Stat. 9(F07), 74–82 (2007)

    MathSciNet  Google Scholar 

  48. Zhang, Y., Zhou, Y.: Symmetries and conserved quantities for fractional action-like Pfaffian variational problems. Nonlinear Dyn. 73(1–2), 783–793 (2013)

    Article  MATH  Google Scholar 

  49. Long, Z.X., Zhang, Y.: Noether’s theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space. Acta Mech. 225(1), 77–90 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  50. Long, Z.X., Zhang, Y.: Fractional Noether theorem based on extended exponentially fractional integral. Int. J. Theor. Phys. 53(3), 841–855 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  51. Zhang, Y.: Perturbation to Noether symmetries and adiabatic invariants for nonconservative dynamic systems. Acta Phys. Sin. 62(16), 164501 (2013). (in Chinese)

    Google Scholar 

  52. Zhang, Y.: Noether symmetries and conserved quantities for fractional action-like variational problems in phase space. Acta. Sci. Nat. Univ. Sunyatsen. 52(4), 45–50 (2013). (in Chinese)

    MATH  Google Scholar 

  53. Birkhoff, G.D.: Dynamical Systems. AMS College Publication, Providence (1927)

    MATH  Google Scholar 

  54. Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, New York (1983)

    Book  MATH  Google Scholar 

  55. Mei, F.X., Shi, R.C., Zhang, Y.F., Wu, H.B.: Dynamics of Birkhoffian System. Beijing Institute of Technology Press, Beijing (1996). (in Chinese)

  56. Galiullan, A.S.: Analytical Dynamics. Nauka, Moscow (1989). (in Russian)

    Google Scholar 

  57. Galiullin, A.S., Gafarov, G.G., Malaishka, R.P., Khwan, A.M.: Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems. UFN, Moscow (1997). (in Russian)

    Google Scholar 

  58. Mei, F.X.: Noether theory of Birkhoffian system. Sci. China (Ser. A) 36(12), 1456–1467 (1993)

    MATH  MathSciNet  Google Scholar 

  59. Mei, F.X.: On the Birkhoffian mechanics. Int. J. Non-Linear Mech. 36(5), 817–834 (2001)

    Article  MATH  Google Scholar 

  60. Guo, Y.X., Luo, S.K., Shang, M., Mei, F.X.: Birkhoffian formulations of nonholonomic constrained systems. Rep. Math. Phys. 47(3), 313–322 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  61. Zheng, G.H., Chen, X.W., Mei, F.X.: First integrals and reduction of the Birkhoffian system. J. Beijing Int. Technol. 10(1), 17–22 (2001)

    MATH  MathSciNet  Google Scholar 

  62. Zhang, Y.: Poisson theory and integration method of Birkhoffian systems in the event space. Chin. Phys. B 19(8), 080301 (2010)

    Article  Google Scholar 

  63. Wu, H.B., Mei, F.X.: Type of integral and reduction for a generalized Birkhoffian system. Chin. Phys. B 20(10), 104501 (2011)

    Article  Google Scholar 

  64. Jiang, W., Li, L., Li, Z.J., Luo, S.K.: Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems. Nonlinear Dyn. 67(2), 1075–1081 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  65. Li, Z., Luo, S.: A new Lie symmetrical method of finding conserved quantity for Birkhoffian systems. Nonlinear Dyn. 70(2), 1117–1124 (2012)

    Article  MATH  Google Scholar 

  66. Zhang, Y., Mei, F.X.: Effects of constraints on Noether symmetries and conserved quantities in a Birkhoffian system. Acta Phys. Sin. 53(8), 2419–2423 (2004). (in Chinese)

    MATH  MathSciNet  Google Scholar 

  67. Mei, F.X.: Dynamics of Generalized Birkhoffian Systems. Science Press, Beijing (2013). (in Chinese)

    Google Scholar 

  68. Mei, F.X.: Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems. Science Press, Beijing (1999). (in Chinese)

    Google Scholar 

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 10972151 and 11272227), the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province (No. CXZZ13_0853) and the Innovation Program for Postgraduate of Suzhou University of Science and Technology (No. SKCX13S_051).

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Authors and Affiliations

  1. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215011, People’s Republic of China

    Yi Zhang

  2. College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, People’s Republic of China

    Xiang-Hua Zhai

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  1. Yi Zhang
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  2. Xiang-Hua Zhai
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Correspondence to Yi Zhang.

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Zhang, Y., Zhai, XH. Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dyn 81, 469–480 (2015). https://doi.org/10.1007/s11071-015-2005-5

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