Nonlinear Dynamics

, Volume 92, Issue 4, pp 1947–1954 | Cite as

Symmetry theories for canonicalized equations of constrained Hamiltonian system

  • Shan Cao
  • Jing-Li Fu
Original Paper


In this paper, we give a new method to investigate Noether and Lie symmetries of constrained Hamiltonian system. Firstly, we study the canonicalization of constrained Hamiltonian system. Through variable transformation, the old variables are replaced by the new variables, and at this point, the motion equations for the constrained Hamiltonian system have been canonicalized. Then, we follow the usual symmetry method for the constrained mechanical system, and the Noether and Lie symmetries of constrained Hamiltonian system are given. Finally, three examples are presented to illustrate the application of the results.


Constrained Hamiltonian system Canonicalization Noether symmetry Lie symmetry 



This project was supported by the National Natural Science Foundation of China (Grant Nos. 11472247, 11272287) and by the Zhejiang Province Key Science and Technology Innovation Team Project (2013TD18).

Compliance with ethical standards

Conflict of interest

Both authors declare that they have no conflict of interest.


  1. 1.
    Dirac, P.A.M.: Lecture on Quantum Mechanics. Yeshiva University Press, New York (1964)zbMATHGoogle Scholar
  2. 2.
    Li, Z.P., Li, A.M.: Quantum Symmetry Properties of Constrained System. Beijing Polytechnic University Press, Beijing (2011)Google Scholar
  3. 3.
    Li, Z.P.: Constrained Hamiltonian System and Their Symmetrical Properties. Beijing Polytechnic University Press, Beijing (1999). (in Chinese)Google Scholar
  4. 4.
    Li, Z.P., Jiang, J.H.: Symmetries in Constrained Canonical System. Science Press, Beijing (2002)Google Scholar
  5. 5.
    Holod, I., Lin, Z.: Verification of electromagnetic fluid-kinetic hybrid electron model in global gyrokinetic particle simulation. Phys. Plasmas 20(3), 1291–475 (2013)CrossRefGoogle Scholar
  6. 6.
    Zhao, Y.Y.: Symmetry and Invariants of Mechanical Systems. Science Press, Beijing (1999)Google Scholar
  7. 7.
    Liu, D.: Noether theorem and inverse theorem for nonholonomic nonconservative dynamical systems. Sci. China Ser. A 4, 37–47 (1991)Google Scholar
  8. 8.
    Mei, F.X.: Noether theory of Birkhoff system. Sci. China 7, 709–717 (1993)Google Scholar
  9. 9.
    Mei, F.X.: Application of Lie Groups and Lie Algebras to Constrained Mechanical Systems, p. 374, 378. Science Press, Beijing (1999). (in Chinese)Google Scholar
  10. 10.
    Dorodnitsyn, V., Kozlov, R., Winternitz, P.: Continuous symmetries of Lagrangians and exact solutions of discrete equations. J. Math. Phys. 45, 336–359 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhang, H.B., Chen, L.Q., Liu, R.W.: First integrals of the discrete nonconservative and nonholonomic systems. Chin. Phys. 14(2), 238–243 (2005)CrossRefGoogle Scholar
  12. 12.
    Fu, J.L., Chen, B.Y., Chen, L.Q.: Noether symmetries of discrete nonholonomic dynamical systems. Phys. Lett. A 373, 409–412 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fu, J.L., Chen, B.Y., Fu, H., et al.: Velocity-dependent symmetries and non-Noether conserved quantities of electromechanical systems. Sci. China Phys. Mech. Astron. 54(2), 288C295 (2011)CrossRefGoogle Scholar
  14. 14.
    Fu, J.L., Li, X.W., Li, C.R., et al.: Symmetries and exact solutions of discrete nonconservative systems. Sci. China Phys. Mech. Astron. 53(9), 1699C1706 (2010)Google Scholar
  15. 15.
    Feng, K., Qin, M.: Symplectic Geometric Algorithms for Hamiltonian Systems. Zhejiang Science and Technology Publishing House, Zhejiang (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Li, Z.P.: Classical and Quantal Dynamics of Constrained System and Their Symmetrical Properties, pp. 163–165. Beijing Polytechnic University Press, Beijing (1993)Google Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematical PhysicsZhejiang Sci-Tech UniversityHangzhouChina

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