Improvement of field method and its application to integrating motion equation in Riemann-Cartan space

Wang Yong; Mei Feng-Xiang; Cao Hui-Ying; Guo Yong-Xin

doi:10.7498/aps.67.20171583

Abstract

Like the Hamilton-Jacobi method, the Vujanović field method transforms the problem of seeking the particular solution of an ordinary differential equations into the problem of finding the complete solution of a first order quasilinear partial differential equation, which is usually called the basic partial differential equation. Due to no need of the strong restrictive conditions required in the classic Hamilton-Jacobi method, the Vujanović field method may be used in many fields, such as non-conservative systems, nonholonomic systems, Birkhoff systems, controllable mechanical systems, etc. Even so, there is still a fundamental difficulty in the Vujanović field method. That is, for most of dynamical systems, it is hard to find the complete solution of the basic partial differential equation. In this paper, the Vujanović field method is improved into a new field method. The purpose of the improved field method is to find the first integrals of the motion equations, but not the particular solutions of the motion equations. The improved field method points out that for a basic partial differential equation with n independent variables, m (m n) first integrals of a dynamical system can be found as long as a solution with m arbitrary constants of the basic partial differential equation is found. In particular, if the complete solution (the complete solution is a special case of m=n) of the basic partial differential equation is found, all first integrals of the dynamical system can be found. That means that the motion of the dynamical system is completely determined. The Vujanović field method is just equivalent to this particular case. The improved field method expands the applicability of the field method, and is simpler than the Vujanović field method. Two examples are given to illustrate the effectiveness of the method. In addition, the improved field method is used to integrate the motion equations in Riemann-Cartan space. For a first-order linear homogenous scleronomous nonholonomic system subjected to an active force, its motion equation in its Riemann-Cartan configuration space can be obtained by a first order nonlinear nonholonomic mapping. Since the motion equations in Riemann-Cartan configuration space contain quasi-speeds, they are often considered to be difficult to solve directly. In this paper we give a briefing of how to construct the motion equations of a first order linear nonholonomic constraint system in its Riemann-Cartan configuration space, and how to obtain the first integrals of the motion equations in the Riemann-Cartan configuration space by the improved field method. This is an effective method to study some nonholonomic nonconservative motions.

Keywords:

Authors and contacts

1.
School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;
2.
School of Information Engineering, Guangdong Medical University, Dongguan 523808, China;
3.
College of Physics, Liaoning University, Shenyang 110036, China;
4.
Physics of Medical Imaging Department, Eastern Liaoning University, Dandong 118001, China

Corresponding author: Guo Yong-Xin, yxguo@lnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11772144, 11572145, 11272050, 11572034, 11202090, 11472124) and the Natural Science Foundation of Guangdong Province, China (Grant No. 2015AO30310178).

References

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[24]	Guo Y X, Liu C, Wang Y, Chang P 2010 Sci. China: Phys. Mech. Astron. 53 1707

Cited By

[1]	Rumyantsev V V, Sumbatov A S 1978 ZAMM 58 477
[2]	Vujanović B 1984 Int. J. Non-Linear Mech. 19 383
[3]	Vujanović B 1981 Int. J. Engng. Sci. 19 1739
[4]	Vujanović B 1987 J. Sound Vib. 114 375
[5]	Mei F X 1992 Acta Armam. 13 47 (in Chinese) [梅凤翔 1992 兵工学报 13 47]
[6]	Mei F X 1992 Appl. Math. Mech. 13 165 (in Chinese) [梅凤翔 1992 应用数学和力学 13 165]
[7]	Mei F X 1989 Acta Mech. Sin. 5 260
[8]	Mei F X 2000 Int. J. Non-Linear Mech. 35 229
[9]	Mei F X 1990 Acta Mech. Sin. 6 160
[10]	Luo S K 1995 Appl. Math. Mech. 16 981 (in Chinese) [罗绍凯 1995 应用数学和力学 16 981]
[11]	Zhang Y 1996 J. B. Inst. Technol. 16 36 (in Chinese) [张毅 1996 北京理工大学学报 16 36]
[12]	Chen X W, Luo S K 1998 Appl. Math. Mech. 19 447 (in Chinese) [陈向炜, 罗绍凯 1998 应用数学和力学 19 447]
[13]	Fu J L, Chen L Q, Luo S K, Chen X W, Wang X M 2001 Acta Phys. Sin. 50 2289 (in Chinese) [傅景礼, 陈立群, 罗绍凯, 陈向炜, 王新民 2001 50 2289]
[14]	Luo S K, Guo Y X, Chen X W, Fu J L 2001 Acta Phys. Sin. 50 2049 (in Chinese) [罗绍凯, 郭永新, 陈向炜, 傅景礼 2001 50 2049]
[15]	Abd-El-Latif G M 2004 Appl. Math. Comput. 147 267
[16]	Kovacic I 2005 Acta Mech. Sin. 21 192
[17]	Ge W K 2006 Acta Phys. Sin. 55 10 (in Chinese) [葛伟宽 2006 55 10]
[18]	Zhang Y 2011 J. Southeast Univ. 27 188
[19]	Li Y M, Mei F X 2010 Acta Phys. Sin. 59 5930 (in Chinese) [李彦敏, 梅凤翔 2010 59 5930]
[20]	Wang Y, Guo Y X 2005 Acta Phys. Sin. 54 5517 (in Chinese) [王勇, 郭永新 2005 54 5517]
[21]	Guo Y X, Wang Y, Chee G Y, Mei F X 2005 J. Math. Phys. 46 062902
[22]	Guo Y X, Liu S X, Liu C, Luo S K, Wang Y 2007 J. Math. Phys. 48 082901
[23]	Wang Y, Guo Y X, L Q S, Liu C 2009 Acta Phys. Sin. 58 5142 (in Chinese) [王勇, 郭永新, 吕群松, 刘畅 2009 58 5142]
[24]	Guo Y X, Liu C, Wang Y, Chang P 2010 Sci. China: Phys. Mech. Astron. 53 1707

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Vol. 67, Issue 3 - 2018-02-05

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