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In this paper, we mainly study the simplification and improvement of Santilli's methods in Birkhoffian system, which is a more general type of basic dynamic system. The theories and methods of Birkhoffian dynamics have been used in hadron physics, quantum physics, rotational relativity theory, and fractional dynamics. As is well known, Lagrangian inverse problem, Hamiltonian inverse problem, and Birkhoffian inverse problem are the main objects of the dynamic inverse problems. The results given by Douglas (Douglas J 1941 Trans. Amer. Math. Soc. 50 71) and Havas[Havas P 1957 Nuovo Cimento Suppl. Ser. X5 363] show that only the self-adjoint Newtonian systems can be represented by Lagrange's equations, so the Lagrangian inverse problem is not universal for a holonomic constrained mechanical system. Furthermore, from the equivalence between Lagrange's equation and Hamilton's equation, Hamiltonian inverse problem is not universal. A natural question is then raised:whether there exists a self-adjoint dynamical model whose inverse problem is universal for holonomic constrained mechanical systems, in the field of analytical mechanics.An in-depth study of this issue in the 1980s by R. M. Santilli shows that a universal self-adjoint model exists for a holonomic constrained mechanic system that satisfies the basic conditions of locality, analyticity, and formality. The Birkhoff's equation is a natural extension of the Hamilton's equation, which shows the geometric properties of a nonconservative system as a general symplectic structure. This more general symplectic structure provides the geometry for the study of the non-conservative system preserving structure algorithms. Therefore, it is particularly important to study the problem of the Birkhoffian representation for the holonomic constrained system.For the inverse problem of Birkhoff's dynamics, studied mainly are the condition under which the mechanical systems can be represented by Birkhoff's equations and the construction method of Birkhoff's functions. However, due to the extensiveness and complexity of the holonomic nonconservative system, Birkhoff's dynamical functions do not have so simple construction method as Lagrange function and Hamilton function. The research results of this issue are very few. The existing construction methods are mainly for three constructions proposed by Santilli[Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York:Springer-Verlag) pp25-28], and there are still many technical problems to be solved in the applications of these methods.In order to solve these problems, this article mainly focuses on the following content. First, according to the existence theorem of Cauchy-Kovalevskaya type equations, we prove that the autonomous system always has an autonomous Birkhoffian representation. Second, a more concise method is given to prove that Santilli's second method can be simplified. An equivalent relationship implied in Santilli's third method is found, an improved Santilli's third method is proposed, and the MATLAB programmatic calculation of the method is studied. Finally, the full text is summarized and the results are discussed.
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Keywords:
- Birkhoffian inverse problem /
- Birkhoffian dynamical functions /
- Santilli' /
- s methods
[1] Birkhoff G D 1927 Dynamical Systems (New York: AMS College Publishers Providence, RI, Vol. IX)
[2] Santilli R M 1978 Foundations of Theoretical Mechanics I (New York: Springer-Verlag) pp219-235
[3] Mei F X 2009 Inverse Problems of Dynamics (Beijing: National Defense Industry Press) pp261-263 (in Chinese) [梅凤翔 2009 动力学逆问题 (北京: 国防工业出版社) 第261–263页]
[4] Douglas J 1941 Trans. Amer. Math. Soc. 50 71
[5] Havas P 1957 Nuovo Cimento Suppl. Ser. X5 363
[6] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry. 2nd Edition. (New York: Springer-Verlag) pp181-210
[7] Sarlet W 1982 J. Phys. A 15 1503
[8] Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York: Springer-Verlag) pp25-28
[9] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoff System (Beijing: Beijing Institute of Technology Press) pp8-25 (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴恵彬 1996 Birkhoff系统动力学 (北京: 北京理工大学出版社) 第8–25页]
[10] 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]
[11] Wu H B, Mei F X 2011 Chin. Phys. B 20 290
[12] Luo S K, He J M, Xu Y L 2016 Int. J. Non-Linear Mech. 78 105
[13] Luo S K, Dai Y, Zhang X T, Yang M J 2017 Int. J. Non-Linear Mech. 97 107
[14] Chen X W, Zhang Y, Mei F X 2017 Chin. J. Theor. Appl. Mech. 49 149 (in Chinese) [陈向炜, 张晔, 梅凤翔 2017 力学学报 49 149]
[15] Fu J L, Fu L P, Chen B Y, Sun Y 2016 Phys. Lett. A 380 15
[16] Kong X L, Wu H B 2017 Acta. Phys. Sin. 66 084501 (in Chinese) [孔新雷, 吴惠彬 2017 物理学报 66 084501]
[17] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[18] Liu C, Song D, Liu S X, Guo Y X 2013 Sci. Chin. Tech. Sci. 43 541 (in Chinese) [刘畅, 宋端, 刘世兴, 郭永新 2013 中国科学: 物理学 力学 天文学 43 541]
[19] Feng K, Qin M Z 2003 Symplectic Geometric Algorithms for Hamiltonian Systems (Hangzhou: Zhejiang Science & Technology Press) pp246-258 (in Chinese) [冯康, 秦孟兆 2003 哈密尔顿系统的辛几何算法 (杭州: 浙江科学技术出版社) 第246–258页]
[20] Zhang X W, Wu J K, Zhu H P, Huang K F 2002 Appl. Math. Mech. 9 915 (in Chinese) [张兴武, 武际可, 朱海平, 黄克服 2002 应用数学和力学 9 915]
[21] Sun Y J, Shang Z J 2005 Phys. Lett. A 336 358
[22] Liu S X, Liu C, Guo Y X 2011 Chin. Phys. B 20 034501
[23] Ding G T 2008 Acta. Phys. Sin. 57 7415 (in Chinese) [丁光涛 2008 物理学报 57 7415]
[24] Cui J C, Liao C C, Zhao Z 2016 Acta. Phys. Sin. 65 180201 (in Chinese) [崔金超, 廖翠萃, 赵喆, 刘世兴 2016 物理学报 65 180201]
[25] Cui J C, Song D, Guo Y X 2012 Acta. Phys. Sin. 61 244501 (in Chinese) [崔金超, 宋端, 郭永新 2012 物理学报 61 244501]
[26] Song D, Liu C, Guo Y X 2013 Appl. Math. Mech. 34 995 (in Chinese) [宋端, 刘畅, 郭永新 2013 应用数学和力学 34 995]
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[1] Birkhoff G D 1927 Dynamical Systems (New York: AMS College Publishers Providence, RI, Vol. IX)
[2] Santilli R M 1978 Foundations of Theoretical Mechanics I (New York: Springer-Verlag) pp219-235
[3] Mei F X 2009 Inverse Problems of Dynamics (Beijing: National Defense Industry Press) pp261-263 (in Chinese) [梅凤翔 2009 动力学逆问题 (北京: 国防工业出版社) 第261–263页]
[4] Douglas J 1941 Trans. Amer. Math. Soc. 50 71
[5] Havas P 1957 Nuovo Cimento Suppl. Ser. X5 363
[6] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry. 2nd Edition. (New York: Springer-Verlag) pp181-210
[7] Sarlet W 1982 J. Phys. A 15 1503
[8] Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York: Springer-Verlag) pp25-28
[9] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoff System (Beijing: Beijing Institute of Technology Press) pp8-25 (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴恵彬 1996 Birkhoff系统动力学 (北京: 北京理工大学出版社) 第8–25页]
[10] 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]
[11] Wu H B, Mei F X 2011 Chin. Phys. B 20 290
[12] Luo S K, He J M, Xu Y L 2016 Int. J. Non-Linear Mech. 78 105
[13] Luo S K, Dai Y, Zhang X T, Yang M J 2017 Int. J. Non-Linear Mech. 97 107
[14] Chen X W, Zhang Y, Mei F X 2017 Chin. J. Theor. Appl. Mech. 49 149 (in Chinese) [陈向炜, 张晔, 梅凤翔 2017 力学学报 49 149]
[15] Fu J L, Fu L P, Chen B Y, Sun Y 2016 Phys. Lett. A 380 15
[16] Kong X L, Wu H B 2017 Acta. Phys. Sin. 66 084501 (in Chinese) [孔新雷, 吴惠彬 2017 物理学报 66 084501]
[17] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[18] Liu C, Song D, Liu S X, Guo Y X 2013 Sci. Chin. Tech. Sci. 43 541 (in Chinese) [刘畅, 宋端, 刘世兴, 郭永新 2013 中国科学: 物理学 力学 天文学 43 541]
[19] Feng K, Qin M Z 2003 Symplectic Geometric Algorithms for Hamiltonian Systems (Hangzhou: Zhejiang Science & Technology Press) pp246-258 (in Chinese) [冯康, 秦孟兆 2003 哈密尔顿系统的辛几何算法 (杭州: 浙江科学技术出版社) 第246–258页]
[20] Zhang X W, Wu J K, Zhu H P, Huang K F 2002 Appl. Math. Mech. 9 915 (in Chinese) [张兴武, 武际可, 朱海平, 黄克服 2002 应用数学和力学 9 915]
[21] Sun Y J, Shang Z J 2005 Phys. Lett. A 336 358
[22] Liu S X, Liu C, Guo Y X 2011 Chin. Phys. B 20 034501
[23] Ding G T 2008 Acta. Phys. Sin. 57 7415 (in Chinese) [丁光涛 2008 物理学报 57 7415]
[24] Cui J C, Liao C C, Zhao Z 2016 Acta. Phys. Sin. 65 180201 (in Chinese) [崔金超, 廖翠萃, 赵喆, 刘世兴 2016 物理学报 65 180201]
[25] Cui J C, Song D, Guo Y X 2012 Acta. Phys. Sin. 61 244501 (in Chinese) [崔金超, 宋端, 郭永新 2012 物理学报 61 244501]
[26] Song D, Liu C, Guo Y X 2013 Appl. Math. Mech. 34 995 (in Chinese) [宋端, 刘畅, 郭永新 2013 应用数学和力学 34 995]
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