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Abstract

In this paper, the Lie symmetry theory of singular mechanico-electrical coupling systems is studied in phase space. Firstly, the Hamilton canonical equation of singular mechanico-electrical coupling systems is established through energy; Secondly, considering the inherent constraints brought by singularity, the determining equations, definitions, structural equations and conserved quantities of Lie symmetry are all given; Finally, an application example is given. It is shown that if the structural equation of Lie symmetry is just right the criterion equation for Noether symmetry, then the Lie symmetry can lead to the conserved quantity of Noether type; if the structural equation of Lie symmetry does not involve the Hamiltonian function, the conserved quantity of non-Noether type can be obtained only through finding the invariant of symmetric group.

Keywords

Singular mechanico-electrical coupling systems Lie symmetries structural equations conserved quantities

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References

Z.P.

, Constrained Hamiltonian System and Its Symmetric Properties, Beijing University of Technology Press, 1999, pp. 25–38.

Dirac

P.A.M.

, Lecture on Quantum Mechanics, Yeshiva University Press, 1964, pp. 67–82.

Dirac

P.A.M.

, Quantization of singular systems, Can J Math 2 (1950), 122–199.

Z.P.

and Jiang

J.H.

, Symmetries in Constrained Canonical System, Science Press, 2002, pp. 15–26.

Z.P.

, The regular form’s generalized Noether theorem of nonholonomic singular systems and its inverse theorem, HuangHuai Journal 3 (1992), 8–16.

Z.P.

, Noether’s theorem of regular form and its application, Science Bulletin 36 (1991), 954–958.

Z.P.

, Classical and Quantum Symmetry Properties of Constrained Systems, Beijing University of Technology Press, 1993, pp. 67–73.

Mei

F.X.

, The Application of Liegroup and Lie Algebra to the Constrained Mechanical Systems, Science Press, 1999, pp. 45–58.

Zhang

and Xue

, Lie symmetries of constraint Hamiltonian system with the second type of constraints, Acta Phys Sinica 50 (2001), 816–819, doi:10.7498/aps.50.816.

10.

Luo

S.K.

, Mei symmetries, Noether symmetries and Lie symmetries of Hamiltonian canonical equations of singular systems, Acta Phys Sinica 53 (2004), 5–11, doi:10.7498/aps.52.2941.

11.

A.M.

and Jiang

J.H.

, Regular symmetry of nonholonomic constrained systems, Journal of Wuhan Technical College of Communications 15 (2013), 1–3.

12.

Zheng

M.L.

, Perturbation and adiabatic invariants of Mei symmetry for constrained Hamilton system, Journal of Yanbian University (Natural Sclence) 43 (2017), 328–335, doi:10.16379/j.cnki.issn.1004-4353.2017.04.008.

13.

J.L.

Chen

X.W.

and Luo

S.K.

, Lie symmetry and conserved quantity of Lagrange–Maxwell system, Acta Mechanica Solida Sinica 21 (2000), 157–160, doi:10.19636/j.cnki.cjsm42-1250/o3.2000.02.010.

14.

Qiu

J.J.

, Electromechanical Analytical Dynamics, Science Press, 1992, pp. 14–25.

15.

Liu

X.W.

and Li

Y.C.

, The Lie-Mei symmetry of electromechanical systems, Journal of Suzhou University of Science and Technology (Natural Sclence) 28 (2011), 15–19.

16.

Y.C.

Xia

L.L.

and Wang

X.M.

, Unified symmetry of mechanico-electrical systems with nonholonomic constraints of non-Chetaev’s type, Acta Phys Sinica 59 (2009), 6733–6737, doi:10.7498/aps.58.6732.

17.

Y.C.

Xia

L.L.

and Liu

, Unified symmetry of mechano-electrical systems with nonholonomic constraints, Chin. Phys. B 17 (2008), 1545–1549, doi:10.1088/1674-1056/17/5/002.

18.

F.F.

, Symmetry and conserved quantity of controllable nonholonomic systems and electromechanical systems, Master Dissertation, Qingdao University, 2006.

19.

Jia

, Application of discrete variational principle in electromechanical coupling systems, Master Dissertation, Liaoning University, 2012.

Lie symmetries and conserved quantities of the singular mechanico-electrical coupling systems

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