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The approximate Noether symmetries and conservation laws for approximate Birkhoffian systems

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Abstract

In this paper, the approximate Noether theorems for approximate Birkhoffian systems are presented and discussed. The approximate Birkhoff equations for the systems are established. The Noether identities for approximate Birkhoffian systems are given, which based upon the Noether symmetry and quasi-symmetry, and the relationship between the approximate Noether symmetries and approximate conservation laws for the systems are established, and the approximate Noether theorems are obtained. The results show that the results under the approximate Hamiltonian systems are a special cases of the approximate Birkhoffian systems, while the results under the approximate Lagrangian systems is equivalent to that under the approximate Hamiltonian systems. Finally, two examples are given to illustrate the application of the results.

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References

  1. Govinder, K.S., Heil, T.G., Uzer, T.: Approximate Noether symmetries. Phys. Lett. A 240(3), 127–131 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Ünal, G.: Approximate generalized symmetries, normal forms and approximate first integrals. Phys. Lett. A 269(1), 13–30 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Feroze, T., Kara, A.H.: Group theoretic methods for approximate invariants and Lagrangians for some classes of y″+εF(t)y′+y = f(y,y′). Int. J. Nonlinear Mech. 37(2), 275–280 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Kara, A.H., Mahomed, F.M., et al.: Partial Noether operators and first integrals via partial Lagrangians. Math. Methods Appl. Sci. 30(16), 2079–2089 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Naeem, I., Mahomed, F.M.: Approximate partial Noether operators and first integrals for coupled nonlinear oscillators. Nonlinear Dyn. 57(1–2), 303–311 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Naeem, I., Mahomed, F.M.: Approximate first integrals for a system of two coupled van der Pol oscillators with linear diffusive coupling. Math. Comput. Appl. 15(4), 720–731 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Naz, R., Naeem, I.: Generalization of approximate partial Noether approach in phase-space. Nonlinear Dyn. 88(1), 735–748 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Naz, R., Naeem, I.: The approximate Noether symmetries and approximate first integrals for the approximate Hamiltonian systems. Nonlinear Dyn. 96(4), 2225–2239 (2019)

    MATH  Google Scholar 

  9. Nass, A.M., Mpungu, K.: Lie symmetry reductions and integrability of approximated small delay stochastic differential equations. Afr. Mat. 32, 199–209 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Lou, Z.M.: Approximate Lie symmetries and approximate invariants of the orbit differential equation for perturbed Kepler system. Acta Phys. Sin. 59(10), 6764–6769 (2010). (in Chinese)

    Google Scholar 

  11. Lou, Z.M.: A new method to obtain first order approximate conserved quantities of second-ordinary dynamics system containing nonlinear perturbation terms. Acta Phys. Sin. 63(6), 060202 (2014). (in Chinese)

    Google Scholar 

  12. Lou, Z.M.: Second order approximate conserved quantities of two dimensional perturbed mechanics system. J. Dyn. Control 13(5), 165–169 (2015). (in Chinese)

    Google Scholar 

  13. Lou, Z.M., Mei, F.X., Chen, Z.D.: The first-order approximate Lie symmetries and approximate conserved quantities of the weak nonlinear coupled two-dimensional anisotropic harmonic oscillator. Acta Phys. Sin. 61(11), 110204 (2012). (in Chinese)

    MATH  Google Scholar 

  14. Jiang, W.A., Xia, L.L.: Approximate Birkhoffian formulations for nonconservative systems. Reports. Math. Phys. 81(2), 137–145 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Jiang, W.A., Xia, Z.W., Xia, L.L.: Approximation closure method for Birkhoffian system under random excitations. Int. J. Dyn. Control 6, 398–405 (2018)

    MathSciNet  Google Scholar 

  16. Jia, L.Q., Wang, X.X., et al.: Special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system. Nonlinear Dyn. 69, 1807–1812 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Sun, X.T., Yang, B.C., et al.: Accurate conserved quantity and approximate conserved quantity deduced from Noether symmetry for a weakly Chetaev nonholonomic system. Nonlinear Dyn. 83, 1563–1568 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Zhang, Y.: Noether quasi-symmetry and approximate Noether conservation laws for weakly nonlinear dynamical equations. Chin. J. Theor. Appl. Mech. 52(6), 1765–1773 (2020). (in Chinese)

    Google Scholar 

  19. Lukashchuk, Y.S., Saburova, R.D.: Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type. Nonlinear Dyn. 93, 295–305 (2018)

    MATH  Google Scholar 

  20. Salvo, R.D., Gorgone, M., Oliveri, F.: A consistent approach to approximate Lie symmetries of differential equations. Nonlinear Dyn. 91, 371–386 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Jamal, S.: A Study of the approximate singular Lagrangian conditional Noether symmetries and first integrals. Int. J. Geom. Methods Modern Phys. 16(3), 1950033 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Zhang, R.F.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95(4), 3041–3048 (2019)

    MATH  Google Scholar 

  23. Zhang, R.F., Li, M.C., et al.: Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method. Chaos Soliton Fractals 154, 111692 (2022)

    MathSciNet  MATH  Google Scholar 

  24. Shen, J.L., Wu, X.Y.: Periodic-soliton and periodic-type solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation by using BNNM. Nonlinear Dyn. 106(1), 831–840 (2021)

    Google Scholar 

  25. Zhang, R.F., Li, M.C.: Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations. Nonlinear Dyn. 108(1), 521–531 (2022)

    Google Scholar 

  26. Zhang, R.F., Li, M.C., et al.: The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM. Nonlinear Dyn. (2023). https://doi.org/10.1007/-s11071-023-08257-5

    Article  Google Scholar 

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

    MATH  Google Scholar 

  28. Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, Berlin (1983)

    MATH  Google Scholar 

  29. Galiullin, A.S., Gafarov, G.G., et al.: Analytical Dynamics of HELMHOLTZ, Birkhoff and Nambu Systems. Moscow: UFN (1997) (in Russian)

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  33. Luo, S.K., Yan, L.X.: Fractional Birkhoffian mechanics. Acta Mech. 226, 829–844 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Zhai, X.H., Zhang, Y.: Noether symmetries and conserved quantities for Birkhoffian systems with time delay. Nonlinear Dyn. 77, 73–86 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Zhai, X.H., Zhang, Y.: Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dyn. 81, 469–480 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  37. 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(5), 887–897 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Zhang, L.J., Zhang, Y.: Non-standard Birkhoffian dynamics and its Noether’s theorems. Commun. Nonlinear Sci. Numer. Simulat. 91, 105435 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Zhang, Y.: Theory of generalized canonical transformations for Birkhoff systems. Adv. Math. Phys. 2020, 9482356 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Liu, S.X., Li, N., Liu, C.: Discrete variational calculation of Whittaker equation in the Birkhoffian farmework. J. Dyn. Control. 13(4), 246–249 (2015). (in Chinese)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Zhang, Y., Tian, X.: Conservation laws for Birkhoffian systems of Herglotz type. Chin. Phys. B 27(9), 090502 (2018)

    Google Scholar 

  43. Zhang, Y.: Noether’s theorem for a time-delayed Birkhoffian system of Herglotz type. Int. J. Nonlinear Mech. 101, 36–43 (2018)

    Google Scholar 

  44. Zhang, Y., Zhai, X.H.: Generalized canonical transformation for second-order Birkhoffian systems on time scales. Theor. Appl. Mech. Lett. 9(6), 1–5 (2019)

    Google Scholar 

  45. Xu, X.X., Zhang, Y.: A new type of adiabatic invariants for disturbed Birkhoffian system of Herglotz type. Chin. J. Phys. 64, 278–284 (2020)

    MathSciNet  Google Scholar 

  46. Song, C.J., Cheng, Y.: Noether’s theorems for nonshifted dynamic systems on time scales. Appl. Math. Comput. 374, 125086 (2020)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, Y.: Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scale. Acta Phys. Sin. 70(24), 244501 (2021). (in Chinese)

    Google Scholar 

  48. Kong, X.L., Wu, H.B., Mei, F.X.: Variational discretization of constrained Birkhoffian systems. Nonlinear Dyn. 78, 329–339 (2014)

    MathSciNet  MATH  Google Scholar 

  49. Ding, J.J., Zhang, Y.: Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay. Chaos Soliton Fractals 138, 109913 (2020)

    MathSciNet  MATH  Google Scholar 

  50. Zhai, X.H., Zhang, Y.: Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations. Complexity 2020, 1691760 (2020)

    MATH  Google Scholar 

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Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12102241, 12272248 and 11972241).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu, 476000, People’s Republic of China

    Shi-Xin Jin

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

    Yi Zhang

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  1. Shi-Xin Jin
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Jin, SX., Zhang, Y. The approximate Noether symmetries and conservation laws for approximate Birkhoffian systems. Nonlinear Dyn 111, 13235–13243 (2023). https://doi.org/10.1007/s11071-023-08556-x

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