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提出了相对论性力学系统的一种新的对称性, 并给出了此对称性导致的守恒量. 提出了相对论性力学系统的Birkhoff对称性, 即对应于相对论性力学系统的一组Birkhoff动力学函数的运动微分方程的解都满足从另一组Birkhoff动力学函数得到的运动微分方程. 证明了与两组Birkhoff动力学函数分别给出的相对论性Birkhoff方程相关联的系数矩阵的各次幂的迹是系统的一个守恒量, 从而将Currie和Saletan提出的力学系统的等效Lagrange函数定理拓展到了相对论性Birkhoff动力学系统. 给出了两个例子以说明结果的正确性.
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关键词:
- 相对论性力学系统 /
- Birkhoff 对称性 /
- 守恒量
A new symmetry of a relativistic mechanical system is put forward, and the corresponding conserved quantity is given. The new symmetry is defined in such a way that if each solution to the differential equations of motion of a relativistic mechanical system corresponding to a set of Birkhoff's dynamical functions satisfies the differential equations of motion obtained by other set of Birkhoff's dynamical functions and vice versa, then the corresponding invariance is called a symmetry of Birkhoffians. We prove that the coefficient matrix which relates to the relativistic Birkhoff's equations obtained from two sets of Birkhoff's dynamical functions, is such that the trace of all its integer powers is a conserved quantity of the system, and therefore a theorem known for nonsingular equivalent Lagrangians presented by Currie and Saletan is extended to a relativistic Birkhoffian system. Two examples are given to illustrate the application of the results.[1] Currie D G, Saletan E J 1966 J. Math. Phys. 7 967
[2] Hojman S, Harleston H 1981 J. Math. Phys. 22 1414
[3] Hojman S 1984 J. Phys. A: Math. Gen. 17 2399
[4] Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) p111 (in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与守恒量 (北京: 科学出版社) 第111页]
[5] Mei F X, Wu H B 2008 Phys. Lett. A 372 2141
[6] Wu H B, Mei F X 2009 Chin. Phys. B 18 3145
[7] Mei F X, Wu H B 2009 Acta Phys. Sin. 58 5919 (in Chinese) [梅凤翔, 吴惠彬 2009 58 5919]
[8] Zhang Y, Ge W K 2009 Acta Phys. Sin. 58 7447 (in Chinese) [张毅, 葛伟宽 2009 58 7447]
[9] Mei F X, Gang T Q, Xie J F 2006 Chin. Phys. 15 1678
[10] Zhang Y 2009 Acta Phys. Sin. 58 7436 (in Chinese) [张毅 2009 58 7436]
[11] Fu J L, Chen L Q, Chen X W, Luo S K, Wang X M 2001 Acta Phys. Sin. 50 2289 (in Chinese) [傅景礼, 陈立群, 陈向炜, 罗绍凯, 王新民 2001 50 2289]
[12] Yan Y 1998 J. Hunan Business College 5 61 (in Chinese) [鄢茵 1998 湖南商学院学报 5 61]
[13] Luo S K, Guo Y X 2007 Commun. Theor. Phys. (Beijing, China) 47 209
[14] Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag)
[15] Hojman S, Urrutia L F 1981 J. Math. Phys. 22 1896
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[1] Currie D G, Saletan E J 1966 J. Math. Phys. 7 967
[2] Hojman S, Harleston H 1981 J. Math. Phys. 22 1414
[3] Hojman S 1984 J. Phys. A: Math. Gen. 17 2399
[4] Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) p111 (in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与守恒量 (北京: 科学出版社) 第111页]
[5] Mei F X, Wu H B 2008 Phys. Lett. A 372 2141
[6] Wu H B, Mei F X 2009 Chin. Phys. B 18 3145
[7] Mei F X, Wu H B 2009 Acta Phys. Sin. 58 5919 (in Chinese) [梅凤翔, 吴惠彬 2009 58 5919]
[8] Zhang Y, Ge W K 2009 Acta Phys. Sin. 58 7447 (in Chinese) [张毅, 葛伟宽 2009 58 7447]
[9] Mei F X, Gang T Q, Xie J F 2006 Chin. Phys. 15 1678
[10] Zhang Y 2009 Acta Phys. Sin. 58 7436 (in Chinese) [张毅 2009 58 7436]
[11] Fu J L, Chen L Q, Chen X W, Luo S K, Wang X M 2001 Acta Phys. Sin. 50 2289 (in Chinese) [傅景礼, 陈立群, 陈向炜, 罗绍凯, 王新民 2001 50 2289]
[12] Yan Y 1998 J. Hunan Business College 5 61 (in Chinese) [鄢茵 1998 湖南商学院学报 5 61]
[13] Luo S K, Guo Y X 2007 Commun. Theor. Phys. (Beijing, China) 47 209
[14] Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag)
[15] Hojman S, Urrutia L F 1981 J. Math. Phys. 22 1896
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