约束Hamilton系统的积分因子和守恒量及其在场论中的应用

Abstract

Abstract:

Field theory is the most important and difficult part in the study of constrained Hamiltonian systems. In recent years, it has became a hot research area. In this paper, a general method that to construct the conservation laws of field theory system based on the integral factor method is presented. Firstly, the general Hamilton canonical equation of constrained Hamiltonian system is structured. Secondly, the definition about integrating factors is given and the conservation theorem for constrained Hamiltonian systems is established. Thirdly, the general Killing equation of constrained Hamiltonian system is deduced, then the integrating factors of constrained Hamiltonian systems are obtained. Finally, two examples are used to demonstrate the effectiveness of this method. Obviously, compared with Noether symmetry method and Lie symmetry method, the integrating factor method of constrained Hamiltonian system has the advantages of clearing calculation step, lessening restrictive conditions and simplifying operation and so on.

Key words: Field theory system, Constrained Hamilton system, Integrating factor, Self couple field

CLC Number:

O369

Jingrun Zhou,Jingli Fu. Integrating Factors and Conserved Quantities for Constrained Hamilton Systems and Its Applications in Field Theory[J].Acta mathematica scientia,Series A, 2019, 39(1): 38-48.

Trendmd

References 45

1	Wilczek F . Quantum field theory. Review of Modern Physics, 1999, 71: S85- S95 doi: 10.1103/RevModPhys.71.S85
2	Li Z P . Constrained Hamiltonian Systems and Their Symmetric Properties. Beijing: Beijing University of Technology Press, 1999
3	Li Z P . Symmetry in field theory for singular Lagrange with Derivatives of higher order. Acta Mathematica Scientia, 1994, 14 (S1): 30- 39
4	Li Z P . The symmetry transformation of the constrained system with high-order derivatives. Acta Mathematica Scientia, 1985, 5 (4): 30- 39
5	Wang Y L , Zhao D Y . The Symmetry of Constrained Hamilton System and Its Application. Jinan: Shandong People Press, 2012
6	Zhang R L , Liu J D , Tang Y F , et al. Canonicalization and symplectic simulation of gyrocenter dynamics in time-independent magnetic fields. Phys of Plasma, 2014, 21: 2445- 2458
7	Tang Y , Long Y . Formal energy of a symplectic scheme for Hamiltonian systems and its applications. Computers and Mathematics with Applications, 1994, 27: 31- 39
8	Zhang R L , Zhu B B , Tu X B , Zhao Y . Convergence analysis of the formal energies of symplctic methods for Hamiltonian systems. Science China, 2016, 59: 1- 18
9	Xu J K , Tan Z . Classfication of solutions for a class of singular integral system. Acta Mathematica Scientia, 2011, 31 (4): 1449- 1456 doi: 10.1016/S0252-9602(11)60330-5
10	Chen X L , Yang J F . Regularity and symmetry of solutions of integral system. Acta Mathematica Scientia, 2012, 32 (5): 1759- 1780 doi: 10.1016/S0252-9602(12)60139-8
11	Qiao Y F , Yue Q W , Dong Y A . Noether conservation laws of holonomic nonconservative dynamical systems in generalized mechanics. Applied Mathematics and Mechanics, 1994, 15: 877- 883 doi: 10.1007/BF02451637
12	Utzky M . Dynamical symmetries and conserved quantities. J Phy A Math Gen, 1979, 12: 973- 981 doi: 10.1088/0305-4470/12/7/012
13	Mei F X , Wu R H , Zhang Y F . Lie symmetries and conserved quantities of nonholonomic systems of non chetaev type. Chinese Journal of Theoretical and Applied Mechanics, 1988, 23: 71- 75
14	Liu R W , Fu J L . Lie symmetries and conserved quantities of non-conservative. Chinese Journal of Theoretical and Applied Mechanics, 1989, 20: 597- 601
15	Han G C , Zhang Y L . Energy equation of generalized mechanical system. Journal of Harbin Engineering University, 2001, 22: 69- 71
16	Zhao Y Y , Mei F X . One symmetry and invariant of dynamical systems. Advances Mechanics, 1993, 23: 360- 372
17	Mei F X . Nonholonomic Mechanics Foundation. Beijing: Beijing Institute of Technology Press, 1985
18	Liu R . Nothers theorem and its inverse of nonholonomic nonconservative dynamical systems. Science in China, 1991, 34A: 37- 47
19	Mei F X . The Application of Ordinary Group and Ordinary Algebra to Constrained Mechanical Systems. Beijing: Science Press, 1999
20	Bluman G W , Kumei S . Symmetries and Differential Equations. Berlin: Springer, 1989
21	Olver P J . Applications of Lie Group to Differential Equations. Berlin: Springer, 2000
22	Olver P J . Equivalence, Invariants and Symmetry. Cambridge: Cambridge University Press, 1995
23	MacCallum M . Differential Equations:Their Solution Using Symmetries. Cambridge: Cambridge University Press, 1989
24	Ovsiannikov L V . Group Analysis of Differential Equations. New York: Academic Press, 1982
25	Gorringe V M , Leach P G L . Lie point symmetries for systems of second order linear ordinary differential equations. Quaestiones Mathematicae, 1988, 11: 95- 117 doi: 10.1080/16073606.1988.9631946
26	Cheh J , Olver P J , Pohjanpelto J . Algorithms for differential invariants of symmetry groups of differential equations. Foundations of Computational Mathematics, 2008, 8: 501- 532 doi: 10.1007/s10208-005-0206-x
27	Olver P J , Pohjanpelto J . Moving frames for Lie pseudo-groups. Canadian Journal of Mathematics, 2008, 60: 1336- 1386
28	Mei J Q , Wang H Y . The generating set of the differential invariant algebra and Maurer-Cartan equations of A (2+1)-dimensional Burgers equation. Journal of Systems Science and Complexity, 2013, 26: 281- 290 doi: 10.1007/s11424-013-2031-7
29	Beffa G . Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces. Proceedings of the American Mathematical Society, 2006, 134: 779- 791 doi: 10.1090/S0002-9939-05-07998-0
30	Olver P J , Pohjanpelto J . Differential invariant algebras of Lie pseudo-groups. Advances in Mathematics, 2009, 222: 1746- 1792 doi: 10.1016/j.aim.2009.06.016
31	Lisle G I , Reid G J . Symmetry classification using noncommutative invariant differential operators. Foundations of Computational Mathematics, 2006, 6: 353- 386 doi: 10.1007/s10208-005-0186-x
32	Zhao Y Y . Lie symmetries and conserved quantities of non-conservative mechanical system (in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 1994, 26: 380- 384
33	Mei F X . Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems. Beijing: Science Press, 1999
34	Mei F X . Lie symmetries and conserved quantities of constrained mechanical systems. Acta Mechanica, 2000, 141: 135- 148 doi: 10.1007/BF01268673
35	Zhang H B . Lie symmetries and conserved quantities of non-holonomic mechanical systems with unilateral Vacco constraints. Chinese Physics, 2002, 11: 1- 4 doi: 10.1088/1009-1963/11/1/301
36	Mei F X . Symmetry and Conserved Quantity of Constrained Mechanical System. Beijing: Beijing Institute of Technology Press, 2004
37	Chen X W , Chang L , Mei F X . Conformal invariance and Hojman conserved quantities of first order Lagrange systems. Chinese Physics B, 2008, 17: 3180- 3184 doi: 10.1088/1674-1056/17/9/004
38	Qiao Y F , Zhang Y L , Zhao S H . Integrating factors and conservation laws for the Raitzins canonical equations of motion of non-conservative dynamical systems. Acta Physica Sinica, 2008, 51: 1661- 1665
39	Zhang Y , Ge W K . Integrating factors and conservation laws for non-holonomic dynamical systems. Acta Physica Sinica, 2003, 52: 2364- 2367
40	Shu F P , Zhang Y . Integrating factors and conservation laws for a generalized Birkhoffian systems. Journal of Huazhong Normal University (Nat Sci), 2014, 48: 42- 45
41	Zheng S W , Qiao Y F . Integrating factors and conservation theorems of Lagrange equations for generalized non-conservative systems in terms of quasi-coordinates. Acta Physica Sinica-China Edition, 2006, 55: 3241- 3245
42	Ibragimov N H . Integrating factors, adjoint equations and Lagrangians. Journal of Mathematical Analysis and Applications, 2006, 318: 742- 757 doi: 10.1016/j.jmaa.2005.11.012
43	Anco S C , Bluman G . Integrating factors and first integrals for ordinary differential equations. European Journal of Applied Mathematics, 1998, 9: 245- 259 doi: 10.1017/S0956792598003477
44	Muriel C , Romeroj I L . First integrals, integrating factors and symmetries of second order differential equation. Journal of Physics A Mathematical and Theoretical, 2009, 42: 2285- 2289
45	Choudhury A G , Guha G , Khanra B . On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlece-Gambler classification. Journal of Mathematical Analysis and Applications, 2009, 360: 651- 664 doi: 10.1016/j.jmaa.2009.06.052

Integrating Factors and Conserved Quantities for Constrained Hamilton Systems and Its Applications in Field Theory

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 45

Related Articles 0

Metrics

Comments

Recommended 10