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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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