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As is well known, the development of analysis mechanics from Lagrangian systems to Birkhoffian systems, achieved the self-adjointness representations of the constrained mechanical systems. Based on the Cauchy-Kovalevsky theorem of the integrability conditions for partial differential equations and the converse of the Poincar lemma, it can be proved that there exists a direct universality of Birkhoff's equations for local Newtonian system by reducing Newton's equations into a first-order form, which means that all local, analytic, regular, finite-dimensional, unconstrained or holonomic, conservative or non-conservative forms always admit, in a star-shaped neighborhood of a regular point of their variables, a representation in terms of first-order Birkhoff's equations in the coordinate and time variables of the experiment. The systems whose equations of motion are represented by the first-order Birkhoff's equations on a symplectic or a contact manifold spanned by the physical variables, are called Birkhoffian systems. The theory and method of Birkhoffian dynamics are used in hadron physics, quantum physics, relativity, rotational relativity, and fractional-order dynamics. At present, for a given dynamical system, it is important and essential to determine whether a Birkhoffian function is the first integral of the system. Although the numerical approximation is an important method of solving the differential equations, the direct theoretical analysis is more helpful for refining the general integral method, and more consistent with the usual way of solving problems of analysis mechanics. In this paper, we study how to judge whether a given Birkhoffian dynamical function to be a first integral of Birkhoff's equations, based on the point of Birkhoffian dynamical functions carrying all the informationabout motion of the system, and use the thought of deriving the first integrals of Hamiltonian systems. In Section 2, the normal first-order form and the Birkhoff's equations of the equations of motion of holonomic systems are introduced. In Section 3, we prove that the Birkhoffian function of an autonomous Birkhoffian system must be a first integral, and the Birkhoffian function of a semi-autonomous system must not be a first integral. Moreover, the energy integral, cyclic integral and Hojman integral of the non-autonomous Birkhoffian systems are given. In Section 4, two examples are given to illustrate the applications of the results. In Section 5, the full text is summarized and the results are discussed. It is necessary to point out that the judging method is effective to determine whether a given Birkhoffian functions can be identified to be a first integral of Birkhoff's equations, but other new first integral cannot be found with this method. One possible method of covering the shortage is to obtain other equivalent Birkhoffian functions in terms of isotopic transformations of Birkhoff's equations, and then use our results to seek the new first integral. In addition, we also hope to develop a more direct method of obtaining the first integrals of Birkhoff's equations in the next study.
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Keywords:
- constrained mechanical systems /
- Birkhoff's equations /
- Birkhoffian dynamical functions /
- first integrals
[1] Santilli R M 1978 Foundations of Theoretical Mechanics I (New York:Springer-Verlag) pp219-235
[2] Chen B 2012 Analytical Dynamics (2nd Ed.) (Beijing:Peking University Press) pp5-15 (in Chinese)[陈滨 2012 分析动力学(第二版) (北京:北京大学出版社) 第5–15页]
[3] Mei F X 1985 Foundations of Mechanics of Nonholonomic Systems (Beijing:Beijing Institute of Technology Press) pp6-45 (in Chinese)[梅凤翔 1985 非完整系统力学基础 (北京:北京工业学院出版社) 第6–45 页]
[4] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry 2nd Edition (New York:Springer-Verlag) pp181-210
[5] Mei F X, Shi R C, Zhang Y F et al. 1996 Dynamics of Birkhoff System (Beijing:Beijing Institute of Technology Press) pp8-25 (in Chinese)[梅凤翔, 史荣昌, 张永发 等 1996 Birkhoff系统动力学 (北京:北京理工大学出版社) 第8–25页]
[6] Douglas J 1941 Trans. Amer. Math. Soc. 50 71
[7] Lee H C 1945 Amer. J. Math. 67 321
[8] Bloch A M 2009 Repor. Math. Phys. 63 225
[9] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[10] Wu H B, Mei F X 2012 Chin. Phys. B 21 064501
[11] Xue Y 2012 Mech. Eng. 34 94 (in Chinese)[薛纭 2012 力学与实践 34 94]
[12] Zhang Y, Xue Y 2009 Chin. Quar. Mech. 30 216 (in Chinese)[张毅, 薛纭2009 力学季刊 30 216]
[13] Chen X W, Luo S K, Mei F X 2002 Appl. Math. Mech. 23 47 (in Chinese)[陈向炜, 罗绍凯, 梅凤翔 2002 应用数学和力学 23 47]
[14] Liu S X, Xing Y, Liu C, Guo Y X 2016 Acta Sci. Natur. Univ. Peking 52 592 (in Chinese)[刘世兴, 邢燕, 刘畅, 郭永新 2016 北京大学学报(自然科学版) 52 592]
[15] Liu C, Liu S X, Mei F X, Guo Y X 2008 Acta Phys. Sin. 57 6709 (in Chinese)[刘畅, 刘世兴, 梅凤翔, 郭永新 2008 物理学报 57 6709]
[16] Mei F X, Wu H B, Li Y M, Chen X W 2016 Chin. J. Theor. Appl. Mech. 48 263 (in Chinese)[梅凤翔, 吴惠彬, 李彦敏, 陈向炜 2016 力学学报 48 263]
[17] Guo Y X, Shang M, Luo S K 2003 Appl. Math. Mech. 24 62 (in Chinese)[郭永新, 尚玫, 罗绍凯 2003 应用数学和力学 24 62]
[18] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing:Beijing Institute of Technology Press) pp62-74 (in Chinese)[梅凤翔 2004 约束力学系统的对称性与守恒量. 北京:北京理工大学出版社 第62–74页]
[19] Luo S K, Xu Y L 2015 Acta Mech. 226 829
[20] Zhang Y, Zhou Y 2013 Nonlin. Dyn. 73 783
[21] Chen X W, Mei F X 2000 Mech. Res. Commu. 27 365
[22] Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York:Springer-Verlag) p54
[23] Kong X L 2014 Ph. D. Dissertation (Beijing:Beijing Institute of Technology) (in Chinese)[孔新雷 2014 博士学位论文 (北京:北京理工大学)]
[24] Cui J C 2014 Ph. D. Dissertation (Beijing:Beijing Institute of Technology) (in Chinese)[崔金超 2014 博士学位论文 (北京:北京理工大学)]
[25] Luo S K 2003 Commun. Theor. Phys. 40 133
[26] Luo S K, Fu J L, Chen X W 2001 Acta Phys. Sin. 50 383 (in Chinese)[罗绍凯, 傅景礼, 陈向炜 2001 物理学报 50 383]
[27] Su H L, Qin M Z 2004 Commun. Theor. Phys. 41 329
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[1] Santilli R M 1978 Foundations of Theoretical Mechanics I (New York:Springer-Verlag) pp219-235
[2] Chen B 2012 Analytical Dynamics (2nd Ed.) (Beijing:Peking University Press) pp5-15 (in Chinese)[陈滨 2012 分析动力学(第二版) (北京:北京大学出版社) 第5–15页]
[3] Mei F X 1985 Foundations of Mechanics of Nonholonomic Systems (Beijing:Beijing Institute of Technology Press) pp6-45 (in Chinese)[梅凤翔 1985 非完整系统力学基础 (北京:北京工业学院出版社) 第6–45 页]
[4] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry 2nd Edition (New York:Springer-Verlag) pp181-210
[5] Mei F X, Shi R C, Zhang Y F et al. 1996 Dynamics of Birkhoff System (Beijing:Beijing Institute of Technology Press) pp8-25 (in Chinese)[梅凤翔, 史荣昌, 张永发 等 1996 Birkhoff系统动力学 (北京:北京理工大学出版社) 第8–25页]
[6] Douglas J 1941 Trans. Amer. Math. Soc. 50 71
[7] Lee H C 1945 Amer. J. Math. 67 321
[8] Bloch A M 2009 Repor. Math. Phys. 63 225
[9] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[10] Wu H B, Mei F X 2012 Chin. Phys. B 21 064501
[11] Xue Y 2012 Mech. Eng. 34 94 (in Chinese)[薛纭 2012 力学与实践 34 94]
[12] Zhang Y, Xue Y 2009 Chin. Quar. Mech. 30 216 (in Chinese)[张毅, 薛纭2009 力学季刊 30 216]
[13] Chen X W, Luo S K, Mei F X 2002 Appl. Math. Mech. 23 47 (in Chinese)[陈向炜, 罗绍凯, 梅凤翔 2002 应用数学和力学 23 47]
[14] Liu S X, Xing Y, Liu C, Guo Y X 2016 Acta Sci. Natur. Univ. Peking 52 592 (in Chinese)[刘世兴, 邢燕, 刘畅, 郭永新 2016 北京大学学报(自然科学版) 52 592]
[15] Liu C, Liu S X, Mei F X, Guo Y X 2008 Acta Phys. Sin. 57 6709 (in Chinese)[刘畅, 刘世兴, 梅凤翔, 郭永新 2008 物理学报 57 6709]
[16] Mei F X, Wu H B, Li Y M, Chen X W 2016 Chin. J. Theor. Appl. Mech. 48 263 (in Chinese)[梅凤翔, 吴惠彬, 李彦敏, 陈向炜 2016 力学学报 48 263]
[17] Guo Y X, Shang M, Luo S K 2003 Appl. Math. Mech. 24 62 (in Chinese)[郭永新, 尚玫, 罗绍凯 2003 应用数学和力学 24 62]
[18] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing:Beijing Institute of Technology Press) pp62-74 (in Chinese)[梅凤翔 2004 约束力学系统的对称性与守恒量. 北京:北京理工大学出版社 第62–74页]
[19] Luo S K, Xu Y L 2015 Acta Mech. 226 829
[20] Zhang Y, Zhou Y 2013 Nonlin. Dyn. 73 783
[21] Chen X W, Mei F X 2000 Mech. Res. Commu. 27 365
[22] Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York:Springer-Verlag) p54
[23] Kong X L 2014 Ph. D. Dissertation (Beijing:Beijing Institute of Technology) (in Chinese)[孔新雷 2014 博士学位论文 (北京:北京理工大学)]
[24] Cui J C 2014 Ph. D. Dissertation (Beijing:Beijing Institute of Technology) (in Chinese)[崔金超 2014 博士学位论文 (北京:北京理工大学)]
[25] Luo S K 2003 Commun. Theor. Phys. 40 133
[26] Luo S K, Fu J L, Chen X W 2001 Acta Phys. Sin. 50 383 (in Chinese)[罗绍凯, 傅景礼, 陈向炜 2001 物理学报 50 383]
[27] Su H L, Qin M Z 2004 Commun. Theor. Phys. 41 329
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