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D. Lee Department of Physics, Columbia University, New York, NY 10027, USA 李荫远 研究员, 院士 Prof. Academician Li Yin-Yuan 中国科学院物理研究所, 北京 100190 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 丁肇中 教授, 院士 Prof. Academician Samuel C. C. Ting LEP3, CERN, CH-1211, Geneva 23, Switzerland 杨振宁 教授, 院士 Prof. Academician C. N. Yang Institute for Theoretical Physics, State University of New York, USA 杨福家 教授, 院士 Prof. Academician Yang Fu-Jia 复旦大学物理二系, 上海 200433 Department of Nuclear Physics, Fudan University, Shanghai 200433, China 周光召 研究员, 院士 Prof. Academician Zhou Guang-Zhao (Chou Kuang-Chao) 中国科学技术协会, 北京 100863 China Association for Science and Technology, Beijing 100863, China 王乃彦 研究员, 院士 Prof. Academician Wang Nai-Yan 中国原子能科学研究院, 北京 102413 China Institute of Atomic Energy, Beijing 102413, China 梁敬魁 研究员, 院士 Prof. Academician Liang Jing-Kui 中国科学院物理研究所, 北京 100190 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Continued ) Chin. Phys. B Vol. 22, No. 10 (2013) 104501 A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems∗ Cui Jin-Chao(崔金超)a), Liu Shi-Xing(刘世兴)b), and Song Duan(宋 端)c)† a)School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China b)College of Physics, Liaoning University, Shenyang 110036, China c)Physics of Medical Imaging Department, Eastern Liaoning University, Dandong 118001, China (Received 21 January 2013; revised manuscript received 24 May 2013) The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results. Keywords: autonomous systems, linear autonomous Birkhoff’s equations, non-Hamiltonian systems, Whit- taker’s equations PACS: 45.05.+x, 02.30.Zz, 45.10.–b DOI: 10.1088/1674-1056/22/10/104501 1. Introduction 2. Linear autonomous Birkhoff equations The Birkhoffian mechanics is the modern development of Birkhoff’s equations can be derived from a linear first- analytic mechanics. We can realize a direct universal rep- order variational principle in the same measure as that of resentation of the inverse problem in terms of the Birkhoff Hamilton’s equations, although in the most general way equations.[1,2] Through comparing linear autonomous Birk- possible.[2] The form of the general Birkhoff equation is hoff equations with nonautonomous Birkhoff equations, we ¶R (t;a) ¶Rm (t;a) can find that the former have a simpler structure and are easier n − a˙n ¶am ¶an to solve, and so they are the basis for investigating Birkhoff’s ¶B(t;a) ¶Rm (t;a) equations, such as the constructive methods of Birkhoffian dy- − + = 0; ¶am ¶t namical functions, integral theory, and structure preserving (m;n = 1;:::;2n); (1) algorithms.[3–9] The well-known Whittaker and Hojman-Urrutia equa- where B is the Birkhoffian usually taken as the energy function [10] tions represent a kind of system with a simple first-order of the system, Rm are a set of Birkhoffian functions. form, but without the traditional Hamiltonian representations. Birkhoff’s equations (1) are said to be autonomous when Thus they are of the linear autonomous non-Hamiltonian type. Rm and B functions do not depend explicitly on time, in which When searching for a Birkhoffian representation of such a sys- case the equations are assumed to be in the simplified form, tem, we naturally expect that the corresponding Birkhoff equa- i.e., tions are also the linear autonomous one. This contributes ¶B(a) to both the integral theory and the algorithm realization for W (a)a˙n − = 0; (m;n = 1;:::;2n); (2) mn ¶am the Birkhoffian formulation of the systems.[11–19] Therefore, it becomes an important problem to determine whether a given where system can be represented via the linear autonomous Birkhoff ¶R ¶R W = n − m (3) equations. This question will be discussed in this paper. mn ¶am ¶an This paper is arranged as follows. In Section 2, starting is called Birkhoff’s tensor, which is an antisymmetrical tensor from the normal first-order form of system, we define the spe- and verifies the Jacobi law, i.e., cific form of the linear autonomous Birkhoff equations. In Section 3, a necessary and sufficient condition is obtained via Wmn + Wnm = 0; (4a) the undetermined tensor method. Then, the methods of con- ¶Wmn ¶Wnt ¶Wtm structing the Birkhoffian and Birkhoffian functions are given. + + = 0: (4b) ¶at ¶am ¶an ∗Project supported by the National Natural Science Foundation of China (Grant Nos. 10932002, 11172120, and 11202090). †Corresponding author. E-mail: [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 104501-1 Chin. Phys. B Vol. 22, No. 10 (2013) 104501 In addition, the regularity condition det(W mn ) 6= 0 means that 3. A condition for transforming an autonomous the 2n equations (2) are independent and can be transformed system into linear autonomous Birkhoff equa- into the equivalent form tions ¶B(a) a˙m = W mn (a) ; (m;n = 1;:::;2n); (5) The authors have expatiated on the undetermined tensor ¶an method in Ref. [8], in which the problem for constructing the mn = −1 where W Wmn . Birkhoffian representation of a nonautonomous system, i.e., Suppose that a given linear autonomous system is de- scribed by first-order ordinary differential equations a˙m = X m (t;a); (m = 1;:::;2n) (13) m m t m m a˙ = At a + a ; a = const:; (m;t = 1;:::;2n); (6) is transformed into solving a first-order linear partial differen- where Am is the constant coefficient such that detAm 6= 0. t t tial equation Now, we shall discuss the forms of linear autonomous Birkhoff equations by combining the physical meaning of the Birkhof- ¶W ¶W mn + mn X r (t;a) fian B. Obviously, if Eq. (5) is equivalent to Eq. (6), we have ¶t ¶ar r (t;a) r (t;a) ¶B(a) m ¶X ¶X W mn = A at + a m ; (m;n;t = 1;:::;2n); (7) + Wmr − Wnr = 0; ¶an t ¶an ¶am or rewrite it as (m;n;r = 1;:::;2n); (14) ¶B(a) m t m = Wmn At a + a ; (m;n;t = 1;:::;2n): (8) ¶an where Wmn is the unknown function. Note that the Birkhoffian B is usually taken as the energy func- When the nonautonomous system (13) degenerates into tion, which means that it should be a quadratic function in an autonomous system variable a. Then the tensor Wmn should be a constant when equation (7) or (8) is verified. Therefore, we restrict the form a˙m = X m (a); (m = 1;:::;2n); (15) of the linear autonomous Birkhoff equations as follows: the tensor Wmn is constant in Eq. (2) or Eq. (5), and the function equation (15) degenerates into the following equation B is in the form of r r ¶Wmn r ¶X (a) ¶X (a) 1 n t n X (a) + W − W = 0; B = a P a + b a ; b = const:; (n;t = 1;:::;2n); (9) r mr n nr m 2 nt n n ¶a ¶a ¶a (m;n;r = 1;:::;2n): (16) where the coefficients Pnt and bn are constants, and satisfy the following conditions We obtain Wmn after solving Eq.