约束Hamilton系统的积分因子和守恒量及其在场论中的应用
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摘要
在约束Hamilton系统的研究中,场论系统一直是重要且难度大的一部分.近年来,场论系统已经成为一个热门的研究领域.论文基于积分因子方法给出了构造场论系统守恒量的一般性方法.首先,构造了约束Hamilton系统的广义Hamilton正则方程;其次,给出了场论系统积分因子的定义和守恒定理;然后,建立了场论系统的广义Killing方程,从而导出系统的积分因子和守恒量;最后,给出了几个场论中的例子以说明这种方法的可行性和有效性.显然,与Noether对称性理论和Lie对称性理论相比较,这种方法具有步骤清晰,计算简便,限制条件少等优点.
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.
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.
引文
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[9] Xu J K, Tan Z. Classfication of solutions for a class of singular integral system. Acta Mathematica Scientia,2011, 31(4):1449-1456
[10] Chen X L,Yang J F. Regularity and symmetry of solutions of integral system. Acta Mathematica Scientia,2012, 32(5):1759-1780
[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
[12] Utzky M. Dynamical symmetries and conserved quantities. J Phy A Math Gen, 1979, 12:973-981
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[15] Han G C, Zhang Y L. Energy equation of generalized mechanical system. Journal of Harbin Engineering University, 2001, 22:69-71
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[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
[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
[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
[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
[30] Olver P J, Pohjanpelto J. Differential invariant algebras of Lie pseudo-groups. Advances in Mathematics,2009, 222:1746-1792
[31] Lisle G I, Reid G J. Symmetry classification using noncommutative invariant differential operators. Foundations of Computational Mathematics, 2006, 6:353-386
[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
[35] Zhang H B. Lie symmetries and conserved quantities of non-holonomic mechanical systems with unilateral Vacco constraints. Chinese Physics, 2002, 11:1-4
[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
[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
[43] Anco S C, Bluman G. Integrating factors and first integrals for ordinary differential equations. European Journal of Applied Mathematics, 1998, 9:245-259
[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
[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
[10] Chen X L,Yang J F. Regularity and symmetry of solutions of integral system. Acta Mathematica Scientia,2012, 32(5):1759-1780
[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
[12] Utzky M. Dynamical symmetries and conserved quantities. J Phy A Math Gen, 1979, 12:973-981
[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
[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
[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
[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
[30] Olver P J, Pohjanpelto J. Differential invariant algebras of Lie pseudo-groups. Advances in Mathematics,2009, 222:1746-1792
[31] Lisle G I, Reid G J. Symmetry classification using noncommutative invariant differential operators. Foundations of Computational Mathematics, 2006, 6:353-386
[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
[35] Zhang H B. Lie symmetries and conserved quantities of non-holonomic mechanical systems with unilateral Vacco constraints. Chinese Physics, 2002, 11:1-4
[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
[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
[43] Anco S C, Bluman G. Integrating factors and first integrals for ordinary differential equations. European Journal of Applied Mathematics, 1998, 9:245-259
[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