Skip to main content
SpringerLink
Log in

Noether symmetry and conserved quantity for Hamiltonian system of Herglotz type on time scales

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

This paper deals with Noether symmetry and conserved quantities for the Hamiltonian system of Herglotz type on time scales. Firstly, the variational principle of Herglotz type for the Hamiltonian system on time scales is proposed. According to the Hamilton–Herglotz action for the Hamiltonian system on time scales and the Dubois–Reymond lemma, the Hamilton canonical equations of Herglotz type on time scales are obtained. Then, based upon the invariance of the Hamilton–Herglotz action on time scales under the infinitesimal transformations of a group, the Noether symmetric transformations for the Hamiltonian system are defined on time scales, and the criterion of the symmetry is established. Furthermore, the Noether’s theorem for the Hamiltonian system of Herglotz type on time scales is obtained because of the relationship between the Noether symmetry and conserved quantity. Besides, the Noether conserved quantities for the Hamiltonian system of Herglotz type are given in both the continuous and the discrete cases. Finally, an example is given to illustrate the application of the results.

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. Hilscher, S.: Differential and difference calculus. Nonlinear Anal. 30(5), 2683–2694 (1997)

    Article  MathSciNet  Google Scholar 

  2. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhauser, Boston (2001)

    Book  MATH  Google Scholar 

  3. Agarwal, R., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141(1–2), 1–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bohner, M., Peterson, A.: Advances in Dynamics Equations on Time Scales. Birkhauser, Boston (2003)

    Book  MATH  Google Scholar 

  5. Ahlbrandt, C.D., Morian, C.: Partial differential equations on time scales. J. Comput. Appl. Math. 141(1–2), 35–55 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bohner, M., Guseinov, G.S.: Partial differentiation on time scales. Dyn. Syst. Appl. 13(3), 351–379 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Gong, Y., Xiang, X.: A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. J. Ind. Manag. Optim. 5(1), 1–10 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Li, Y.K., Wang, P.: Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete Contin. Dyn. Syst. (Ser. S) 10(3), 463–473 (2017)

    Article  MathSciNet  Google Scholar 

  9. Jin, S.X., Zhang, Y.: Methods of reduction for Lagrange systems on time scales. Chin. Phys. B 26(1), 014501 (2017)

    Article  Google Scholar 

  10. Zhai, X.H., Zhang, Y.: Noether theorem for non-conservative systems with time delay on time scales. Commun. Nonlinear Sci. Numer. Simul. 52, 32–43 (2017)

    Article  MathSciNet  Google Scholar 

  11. Bohner, M.: Calculus of variation in time scales. Dyn. Syst. Appl. 13(12), 339–349 (2004)

    MATH  Google Scholar 

  12. Ferreira, R.A.C., Malinowska, A.B., Torres, D.F.M.: Optimality conditions for the calculus of variations with higher-order delta derivatives. Appl. Math. Lett. 24(1), 87–92 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bartosiewicz, Z., Martins, N., Torres, D.F.M.: The second Euler–Lagrange equation of variational calculus on time scales. Eur. J. Control 17(1), 9–18 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bartosiewicz, Z., Torres, D.F.M.: Noether’s theorem on time scales. J. Math. Anal. Appl. 342(2), 1220–1226 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Noether, A.E.: Invariante Variationsprobleme. Nachr. Akad. Wiss. Gott. Math. Phys. 2, 235–257 (1918)

    MATH  Google Scholar 

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

    Google Scholar 

  17. Mei, F.X.: Symmetries and Conserved Quantities of Constrained Mechanical Systems. Beijing Institute of Technology Press, Beijing (2004). (in Chinese)

    Google Scholar 

  18. 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)

    MathSciNet  MATH  Google Scholar 

  19. Zhang, Y., Chen, S.: Effects of non-conservative forces and nonholonomic constraints on Noether symmetries of a Hamilton system. J. Univ. Sci. Tech. Suzhou (Nat. Sci.) 22(2), 1–18 (2005)

    MATH  Google Scholar 

  20. 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  MathSciNet  MATH  Google Scholar 

  21. Yan, B., Zhang, Y.: Noether’s theorem for fractional Birkhoffian systems of variable order. Acta Mech. 227(9), 2439–2449 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Djordje, M.: Generalized Noether’s theorem for continuous mechanical systems. Acta Mech. 228(3), 901–917 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Song, J., Zhang, Y.: Noether’s theorems for dynamical systems of two kinds of non-standard Hamiltonians. Acta Mech. 229(1), 285–297 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Malinowska, A.B., Martins, N.: The second Noether theorem on time scales. Abstr. Anal. Appl. 2013, 1728–1749 (2013)

    Article  Google Scholar 

  25. Cai, P.P., Fu, J.L., Guo, Y.X.: Noether symmetries of the nonconservative and nonholonomic systems on time scales. Sci. China Phys. Mech. Astron. 56(5), 1017–1028 (2013)

    Article  Google Scholar 

  26. Malinowska, A.B., Ammi, M.R.S.: Noether’s theorem for control problems on time scales. Int. J. Differ. Equ. 9, 87–100 (2014)

    Google Scholar 

  27. Peng, K.K., Luo, Y.P.: Dynamics symmetries of Hamiltonian system on time scales. J. Math. Phys. 55(4), 60–75 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, Y.: Noether theory for Hamiltonian system on time scales. Chin. Q. Mech. 2, 214–224 (2016)

    Google Scholar 

  29. Song, C.J., Zhang, Y.: Noether theorem for Birkhoffian systems on time scales. J. Math. Phys. 56(10), 102701 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shi, Y.F., Zhang, Y.: Noether theorem for Birkhoffian system on time scales. J. Univ. Sci. Tech. Suzhou (Nat. Sci.) 34(3), 1–7 (2017)

    MATH  Google Scholar 

  31. Zu, Q.H., Zhu, J.Q.: Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales. J. Math. Phys. 57(8), 082701 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Herglotz, G.: Berührungstransformationen. Lectures at the University of Göttingen, Göttingen (1930)

    Google Scholar 

  33. Georgieva, B., Guenther, R.: First Noether-type theorem for the generalized variational principle of Herglotz. Topol. Methods Nonlinear Anal. 20(1), 261–273 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Georgieva, B., Guenther, R., Bodurov, T.: Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem. J. Math. Phys. 44(9), 3911–3927 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Santos, S.P.S., Martins, N., Torres, D.F.M.: Noether’s theorem for higher-order variational problems of Herglotz type. Discrete Contin. Dyn. Syst. 2015, 990–999 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Santos, S.P.S., Martins, N., Torres, D.F.M.: An optimal control approach to Herglotz variational problems. In: Plakhov, A., Tchemisova, T., Freitas, A. (eds.) Optimization in the Natural Sciences, pp. 107–117. Springer, Switzerland (2015)

    Chapter  Google Scholar 

  37. Almeida, R., Malinowska, A.B.: Fractional variational principle of Herglotz. Discrete Contin. Dyn. Syst. (Ser. B) 19(8), 2367–2381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Santos, S.P.S., Martins, N., Torres, D.F.M.: Variational problems of Herglotz type with time delay: Dubois–Reymond condition and Noether’s first theorem. Discrete Contin. Dyn. Syst. (Ser. A) 35(9), 4593–4610 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Santos, S.P.S., Martins, N., Torres, D.F.M.: Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete Contin. Dyn. Syst. (Ser. S) 11(1), 91–102 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, Y.: Variational problem of Herglotz type for Birkhoff system and its Noether’s theorems. Acta Mech. 228(4), 1–12 (2017)

    MathSciNet  Google Scholar 

  41. Tian, X., Zhang, Y.: Noether’s theorem and its inverse of Birkhoffian system in event space based on Herglotz variational problem. Int. J. Theor. Phys. 57, 887–897 (2017). https://doi.org/10.1007/s10773-017-3621-2

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, Y.: Generalized Herglotz variational principle type for nonconservative system in phase space and Noether’s theorem. Chin. J. Theor. Appl. Mech. 48(6), 1382–1389 (2016)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Xue Tian

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

    Yi Zhang

Authors
  1. Xue Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, X., Zhang, Y. Noether symmetry and conserved quantity for Hamiltonian system of Herglotz type on time scales. Acta Mech 229, 3601–3611 (2018). https://doi.org/10.1007/s00707-018-2188-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2188-1

Search

Navigation