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Herglotz-type principle and first integrals for nonholonomic systems in phase space

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Abstract

The Herglotz-type principle and first integrals in phase space for nonconservative nonholonomic systems are researched. Firstly, we focus on Herglotz-type first integrals for generalized Chaplygin systems. For nonconservative generalized Chaplygin systems in phase space, we establish differential variational principle of Herglotz type and derive the transformation of principle’s invariance condition. Then conservation theorem with corresponding inverse theorem of the system is established, and Herglotz-type Noether first integrals are given. Secondly, we investigate Herglotz-type first integrals for general nonholonomic systems and establish the differential variational principle of Herglotz type for general nonholonomic systems in phase space; then, the conservation theorem and corresponding inverse theorem based on principle’s invariance conditions are obtained. Further, Herglotz-type Noether first integrals are presented. Finally, taking generalized Chaplygin system with nonconservative forces and Appell–Hamel problem as examples, their Herglotz Noether first integrals are given by using the theorems we obtained.

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References

  1. Mei, F.X.: Anal. Mech. (2). Beijing Institute of Technology Press, Beijing (2013). (in Chinese)

    Google Scholar 

  2. Herglotz, G.: Gesammelte Schriften. Vandenhoeck & Ruprecht, Göttingen (1979)

    MATH  Google Scholar 

  3. Georgieva, B.: Symmetries of the generalized variational functional of Herglotz for several independent variables. Z. Anal. Anwend. 30(3), 253–268 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Georgieva, B., Bodurov, T.: Identities from infinite-dimensional symmetries of Herglotz variational functional. J. Math. Phys. 54(6), 062901 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Muatjetjeja, B., Khalique, M.C.: Lagrangian approach to a generalized coupled Lane–Emden system: symmetries and first integrals. Commun. Nonlinear Sci. Numer. Simul. 15(5), 1166–1171 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Muatjetjeja, B., Khalique, M.C.: Emden-Fowler type system: Noether symmetries and first integrals. Acta Math. Sci. 32(5), 1959–1966 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Georgieva, B., Guenther, R.: First Noether-type theorem for the generalized variational principle of Herglotz. Topol. Methods Nonlinear Anal. 20(2), 261–273 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Santos, S.P.S., Martins, N., Torres, D.F.M.: Higher-order variational problems of Herglotz type. Vietnam J. Math. 42(4), 409–419 (2014)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  10. Zhang, Y.: Variational problem of Herglotz type for Birkhoffian system and its Noether’s theorems. Acta Mech. 228(4), 1481–1492 (2017)

    MathSciNet  MATH  Google Scholar 

  11. 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.-A. 35(9), 4593–4610 (2015)

    MathSciNet  MATH  Google Scholar 

  12. 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. 11(1), 91–102 (2018)

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

  14. Tavares, D., Almeida, R., Torres, D.F.M.: Fractional Herglotz variational problems of variable order. Discrete Contin. Dyn. Syst. 11(1), 143–154 (2018)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Zhang, Y., Tian, X.: Conservation laws of nonconservative nonholonomic system based on Herglotz variational problem. Phys. Lett. A 383(8), 691–696 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Chen, B.: Analytical Dynamics. Beijing Institute of Technology Press, Beijing (2012). (in Chinese)

    Google Scholar 

  19. Mei, F.X., Wu, H.B., Li, Y.M.: A Brief History of Analytical Mechanics. Science Press, Beijing (2019). (in Chinese)

    Google Scholar 

  20. Meijaard, J.P., Papadopoulos, J.M., Ruina, A., Schwab, A.L.: Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review. P. Roy. Soc. A-Math. Phy. 463(2084), 1955–1982 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Kang, H.K., Liu, C.S., Jia, Y.B.: Inverse dynamics and energy optimal trajectories for a wheeled mobile robot. Int. J. Mech. Sci. 134, 576–588 (2017)

    Google Scholar 

  22. Xiong, J.M., Wang, N.N., Liu, C.S.: Bicycle dynamics and its circular solution on a revolution surface. Acta Mech. Sinica-Prc. 36(1), 220–233 (2020)

    MathSciNet  Google Scholar 

  23. He, X.D., Geng, Z.Y.: Consensus-based formation control for nonholonomic vehicles with parallel desired formations. Int. J. Control 94(2), 507–520 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Wang, B.F., Li, S., Guo, J., Chen, Q.W.: Car-like mobile robot path planning in rough terrain using multi-objective particle swarm optimization algorithm. Neurocomputing 282, 42–51 (2018)

    Google Scholar 

  25. Ye, J.: Hybrid trigonometric compound function neural networks for tracking control of a nonholonomic mobile robot. Intell. Serv. Robot. 7(4), 235–244 (2014)

    Google Scholar 

  26. Pappalardo, C.M., Guida, D.: On the dynamics and control of underactuated nonholonomic mechanical systems and applications to mobile robots. Arch. Appl. Mech. 89(4), 669–698 (2019)

    Google Scholar 

  27. Cen, H., Singh, B.K.: Nonholonomic wheeled mobile robot trajectory tracking control based on improved sliding mode variable structure. Wirel. Commun. Mob. Comut. 2021, 2974839 (2021)

    Google Scholar 

  28. Matveev, A.S., Nikolaev, M.S.: Hybrid control for tracking environmental level sets by nonholonomic robots in maze-like environments. Nonlinear Anal-Hybri. 39, 100982 (2021)

    MathSciNet  MATH  Google Scholar 

  29. Tarasov, V.E., Zaslavsky, G.M.: Nonholonomic constraints with fractional derivatives. J. PHYS. A-MATH. GEN. 39(31), 9797–9815 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Vacaru, S.I.: Fractional nonholonomic Ricci flows. Chaos, Solitons and Fractals 45(9–10), 1266–1276 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Mei, F.X.: Foundations of Mechanics of Nonholonomic Systems. Beijing Institute of Technology Press, Beijing (1985). (in Chinese)

    Google Scholar 

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

    Google Scholar 

  33. Jin, S.X., Zhang, Y.: Methods of reduction for Lagrange systems on time scales with nabla derivatives. Chin. Phys. B 26(1), 247–253 (2017)

    MathSciNet  Google Scholar 

  34. Jin, S.X., Zhang, Y.: Generalized chaplygin equations for non-holonomic systems on time scales. Chin. Phys. B 27(2), 020502 (2018)

    MathSciNet  Google Scholar 

  35. Muatjetjeja, B., Khalique, M.C.: A variational formulation approach to a generalized coupled inhomogeneous Emden-Fowler system. Appl. Anal. 93(3), 466–474 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Shen, H.C.: Routh equation of nonholonomic dynamical systems: from Chetaev condition to Euler condition. Acta Phys. Sin-Ch. Ed. 54(6), 2468–2473 (2005). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  37. Paul, P., Cristian, I.: Nonlinear constraints in nonholonomic mechanics. J. Geom. Mech. 6(4), 527–547 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Muatjetjeja, B., Khalique, M.C.: A variational approach to an inhomogeneous second-order ordinary differential system. Abstr. Appl. Anal. 2013, 197219 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Fang, J.H.: The conservation law of nonholonomic system of second-order non-Chetave’s type in event space. Appl. Math. Mech. 23(1), 89–94 (2002)

    MathSciNet  MATH  Google Scholar 

  40. Mu, X.W., Yu, J.M., Cheng, G.F.: Adaptive regulation of high order nonholonomic systems. Appl. Math. Mech. 27(4), 501–507 (2006)

    MathSciNet  MATH  Google Scholar 

  41. Li, Z.P.: Classical and Quantal Dynamics of Constrained Systems and their Symmetrical Properties. Beijing Polytechnic University Press, Beijing (1993). ((in Chinese))

    Google Scholar 

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

    Google Scholar 

  43. Mei, F.X.: Nonholonomic mechanics. Appl. Mech. Rev. 53(11), 283–305 (2000)

    Google Scholar 

  44. Mei, F.X.: Foundations of Mechanics of Nonholonomic Systems. Beijing Institute of Technology Press, Beijing (1985). (in Chinese)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 12272248 and 11972241) and the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province of China (No. KYCX22_3251).

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Authors and Affiliations

  1. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, 215009, China

    Xin-Chang Dong

  2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215011, China

    Yi Zhang

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Dong, XC., Zhang, Y. Herglotz-type principle and first integrals for nonholonomic systems in phase space. Acta Mech 234, 6083–6095 (2023). https://doi.org/10.1007/s00707-023-03707-y

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