Skip to main content
SpringerLink
Log in

Fractional Birkhoffian mechanics

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff–Birkhoff principle, and give four kinds of fractional Pfaff–Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff–Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman–Urrutia model, the fractional Hénon–Heiles model and the fractional Emden model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    Book  MATH  Google Scholar 

  2. Mei F.X.: Nonholonomic mechanics. ASME Appl. Mech. Rev. 53, 283–305 (2000)

    Article  Google Scholar 

  3. Feng K.: On Difference Schemes and Symplectic Geometry. Science Press, Beijing (1985)

    Google Scholar 

  4. Olver P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)

    Book  MATH  Google Scholar 

  5. Zhu W.Q.: Dynamics and Control of Nonlinear Stochastic System: Hamilton Theory System Frame. Science Press, Beijing (2003)

    Google Scholar 

  6. Luo S.K., Zhang Y.F.: Advances in the Study of Dynamics of Constrained System. Science Press, Beijing (2008)

    Google Scholar 

  7. Luo S.K., Li Z.J., Li L.: A new Lie symmetrical method of finding a conserved quantity for a dynamical system in phase space. Acta Mech. 223, 2621–2632 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cai J.L.: Conformal invariance and conserved quantity of Hamilton system under second-class Mei symmetry. Acta Phys. Pol. A 117, 445–448 (2010)

    Google Scholar 

  9. Jia L.Q., Zheng S.W.: Mei symmetry of generalized Hamilton systems with additional terms. Acta Phys. Sin. 55, 3829–3832 (2006)

    MATH  MathSciNet  Google Scholar 

  10. Luo S.K.: New types of the Lie symmetries and conserved quantities for a relativistic Hamilton system. Chin. Phys. Lett. 20, 597–599 (2003)

    Article  Google Scholar 

  11. Luo S.K.: Mei symmetry, Noether symmetry and Lie symmetry of Hamiltonian systems. Acta Phys. Sin. 52, 2941–2944 (2003)

    MATH  Google Scholar 

  12. Birkhoff G.D.: Dynamical Systems. AMS College Publisher, Providence (1927)

    MATH  Google Scholar 

  13. Santilli R.M.: Foundations of Theoretical Mechanics I. Springer, New York (1978)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Mei F.X., Shi R.C., Zhang Y.F., Wu H.B.: Dynamics of Birkhoff Systems. Beijing Institute of Technology, Beijing (1996)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Chen X.W.: Global Analysis for Birkhoff Systems. Henan University Press, Kaifeng (2002)

    Google Scholar 

  18. Chen X.W., Li Y.M.: Equilibrium points and periodic orbits of higher order autonomous generalized Birkhoff system. Acta Mech. 224, 1593–1599 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Chen X.W.: Closed orbits and limit cycles of second-order autonomous Birkhoff systems. Chin. Phys. 12, 586–589 (2003)

    Article  Google Scholar 

  20. Chen X.W.: Chaos in the second order autonomous Birkhoff system with a heteroclinic circle. Chin. Phys. 11, 441–444 (2002)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Luo S.K.: First integrals and integral invariants of relativistic Birkhoffian systems. Commun. Theor. Phys. 40, 133–136 (2003)

    Article  MATH  Google Scholar 

  23. Luo S.K.: Form invariance and Lie symmetries of rotational relativistic Birkhoff system. Chin. Phys. Lett. 19, 449–451 (2002)

    Article  Google Scholar 

  24. Zhang Y.: A geometrical approach to Hojman theorem of a rotational relativistic Birkhoffian system. Commun. Theor. Phys. 42, 669–671 (2004)

    Article  MATH  Google Scholar 

  25. Su H.L., Qin M.Z.: Symplectic schemes for Birkhoffian system. Commun. Theor. Phys. 41, 329–334 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mandelbrot B.B.: The Fractal Geometry of Nature. W.H. Freeman, New York (1982)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Klimek M.: Fractional sequential mechanics—model with symmetric fractional derivatives. Czech. J. Phys. 51, 1348–1354 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Klimek M.: Stationary–conservation laws for fractional differential equations with variable coefficients. J. Phys. A: Math. Gen. 35, 6675–6693 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. 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 

  33. Cresson J.: Fractional embedding of differential operators and Lagrangian systems. IHÉS, Paris (2006)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  35. Malinowska A.B., Torres D.F.M.: Towards a combined fractional mechanics and quantization. Fract. Calc. Appl. Anal. 15, 407–417 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Odzijewicz T., Malinowska A.B., Torres D.F.M.: Generalized fractional calculus with applications to the calculus of variations. Comput. Math. Appl. 64, 3351–3366 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  38. Tarasov V.E., Zaslavsky G.M.: Nonholonomic constraints with fractional derivatives. J. Phys. A: Math. Gen. 39, 9797–9815 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Tarasov V.E.: Fractional systems and fractional Bogoliubov hierarchy equations. Phys. Rev. E 71, 011102 (2005)

    Article  MathSciNet  Google Scholar 

  40. Luo, S.K., Li, L., Xu, Y.L.: Lie algebraic structure and generalized Poisson conservation law for fractional generalized Hamiltonian systems. Acta Mech. doi:10.1007/s00707-014-1101-9 (2014)

  41. Li L., Luo S.K.: Fractional generalized Hamiltonian mechanics. Acta Mech. 224, 1757–1771 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  42. Luo S.K., Li L.: Fractional generalized Hamiltonian equations and its integral invariants. Nonlinear Dyn. 73, 339–346 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  43. Xu Y.L., Luo S.K.: Stability for manifolds of equilibrium state of fractional generalized Hamiltonian systems. Nonlinear Dyn. 76, 657–672 (2014)

    Article  MathSciNet  Google Scholar 

  44. Baleanu D., Muslih S.I., Rabei E.M.: On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative. Nonlinear Dyn. 53, 67–74 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  45. Jarad F., Abdeljawad T., Baleanu D.: Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn. 62, 609–614 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  46. Chen X.W., Zhao G.L., Mei F.X.: A fractional gradient representation of the Poincaré equations. Nonlinear Dyn. 73, 579–582 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  47. Chen L.Q., Zhao W.J., Zu W.J.: Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. J. Sound Vib. 278, 861–871 (2004)

    Article  MATH  Google Scholar 

  48. Chen L.C., Zhu W.Q.: Stochastic averaging of strongly nonlinear oscillators with small fractional derivative damping under combined harmonic and white noise excitations. Nonlinear Dyn. 56, 231–241 (2009)

    Article  MATH  Google Scholar 

  49. Wang Z.H., Hu H.Y.: Stability of a linear oscillator with damping force of fractional order derivative. Sci. China: Phys. Mech. Astron. 53, 345–352 (2010)

    Article  Google Scholar 

  50. Drozdov A.D., Israel B.: Fractional differential models in finite viscoelasticity. Acta Mech. 124, 155–180 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  51. Ding G.T.: Hamiltonization of Whittaker equations. Acta Phys. Sin. 59, 8326–8329 (2007)

    Google Scholar 

  52. Hojman S., Urrutia L.F.: On the inverse problem of the calculus of variations. J. Math. Phys. 22, 1896–1903 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  53. Ding G.T.: Hamiltonization of Birkhoffian systems. Acta Phys. Sin. 60, 044502 (2011)

    Google Scholar 

  54. Henon M., Helles C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)

    Article  Google Scholar 

  55. Brack M.: Bifurcation cascades and self-similarity of periodic orbits with analytical scaling constants in Hénon–Heiles type potentials. Found. Phys. 31, 209–232 (2001)

    Article  MathSciNet  Google Scholar 

  56. Aguirre J., Vallejo J.C., Sanjua’n M.A.F.: Wada basins and chaotic invariant sets in the Hénon–Heiles system. Phys. Rev. E 64, 066208 (2001)

    Article  Google Scholar 

  57. Gu S.L., Zhang H.B.: Mei symmetry, Noether symmetry and Lie symmetry of an Emden system. Acta Phys. Sin. 55, 5594–5597 (2006)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou, 310018, People’s Republic of China

    Shao-Kai Luo & Yan-Li Xu

Authors
  1. Shao-Kai Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan-Li Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shao-Kai Luo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, SK., Xu, YL. Fractional Birkhoffian mechanics. Acta Mech 226, 829–844 (2015). https://doi.org/10.1007/s00707-014-1230-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-014-1230-1

Keywords

Search

Navigation