一种建立非完整系统运动方程的新方法
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摘要
本文提出了一种不基于任何变分原理而建立非完整系统基本运动方程的新方法.利用动力学方程与广义坐标的选取以及非完整约束函数选取的无关性,结合矩阵的乘积规则,提出了用于表述力学系统协变性的双指标张量分析方法;将力学系统的首次积分等效为作用在系统上的非完整约束,说明了非完整系统的自洽性,进而根据非完整系统的运动方程在首次积分约束下的不变性、非完整约束反力在广义坐标和等效非完整约束函数组变换下的不变性,反推出了非完整约束反力所必须满足的形式,以此建立了非完整系统的基本运动方程.本文提出的方法完全基于非完整系统的自洽性与协变性,不仅没有使用任何先验的D’Alembert-Lagrange, Gauss, Jourdian或Hamilton变分原理,而且还为非完整系统Chetaev条件的成立提供一个合理的解释,并且从自洽性的角度说明了基于Hamilton原理所导出的Vakonomic力学不是非完整系统的合理模型.
In this paper, we present a new method of establishing the fundamental motion equations of nonholonomic systems without using any variational principles. According to the invariance of motion equations under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, the double index tensor analysis method is proposed cooperating with the product rules of matrix. Then we let first integrals equivalent to the corresponding nonholonomic constraints imposed upon the system to explain the self consistency of nonholonomic systems, it is shown that the self consistency is the core of nonholonomic mechanics and can verify which model is appropriate for describing nonholonomic systems. Therefore, the form of nonholonomic constraint force is determined only by three qualities: the invariance of the motion equations under the constraint of first integrals, the invariance of nonholonomic constraint force under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, to establish the fundamental motion equation of nonholonomic systems. Using the double index tensor analysis method, the three qualities and the Moore-Penrose generalized inverse of matrix theory, the equation of motion of nonholonomic systems are derived. Some classical equations in nonholonomic systems, such as the Routh equation, the Nielsen equation and the Chaplygin equation are also derived to prove our method. The method is totally based on the self consistency of nonholonomic systems and the covariation of motion equations, these two characters are naturally derived from mathematical and mechanical requirements of nonholonomic systems. Not only does not use any transcendental variational principles, such as D'Alembert-Lagrange principle, Gauss principle or Jourdian principle, but also illustrate that the validity of the three principles are attributed to the two characters. Further more, this method even provides a reasonable explanation for Chetaev conditions in nonholonomic systems, and shows that the vakonomic mechanics derived by Hamilton's principle is not an appropriate model for nonholonomic systems.
In this paper, we present a new method of establishing the fundamental motion equations of nonholonomic systems without using any variational principles. According to the invariance of motion equations under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, the double index tensor analysis method is proposed cooperating with the product rules of matrix. Then we let first integrals equivalent to the corresponding nonholonomic constraints imposed upon the system to explain the self consistency of nonholonomic systems, it is shown that the self consistency is the core of nonholonomic mechanics and can verify which model is appropriate for describing nonholonomic systems. Therefore, the form of nonholonomic constraint force is determined only by three qualities: the invariance of the motion equations under the constraint of first integrals, the invariance of nonholonomic constraint force under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, to establish the fundamental motion equation of nonholonomic systems. Using the double index tensor analysis method, the three qualities and the Moore-Penrose generalized inverse of matrix theory, the equation of motion of nonholonomic systems are derived. Some classical equations in nonholonomic systems, such as the Routh equation, the Nielsen equation and the Chaplygin equation are also derived to prove our method. The method is totally based on the self consistency of nonholonomic systems and the covariation of motion equations, these two characters are naturally derived from mathematical and mechanical requirements of nonholonomic systems. Not only does not use any transcendental variational principles, such as D'Alembert-Lagrange principle, Gauss principle or Jourdian principle, but also illustrate that the validity of the three principles are attributed to the two characters. Further more, this method even provides a reasonable explanation for Chetaev conditions in nonholonomic systems, and shows that the vakonomic mechanics derived by Hamilton's principle is not an appropriate model for nonholonomic systems.
引文
1 Cendra H, Diaz V A. Lagrange-d’Alembert-Poincare equations by several stages. J Geometric Mech, 2018, 10:1–41
2 Chen B. Analytical Dynamics(in Chinese). 2nd ed. Beijing:Peking University Press, 2012[陈滨.分析动力学.第2版.北京:北京大学出版社,2012]
3 Adamov B I, Kobrin A I. Methods of analytic mechanics in the problem of adaptive identification with constraints. Vestnik Moskov. Univ Ser 1.Mat Mekh, 2016, 5:63–67
4 Liu C S. The fundamental equations in analytical mechanics for nonholonomic systems(in Chinese). Acta Sci Nat Univ Peking, 2016, 52:756–766[刘才山.分析动力学中的基本方程与非完整约束.北京大学学报(自然科学版), 2016, 52:756–766]
5 Song H, Liang L. Investigation of power-type variational principles in liquid-filled system. Appl Math Mech-Engl Ed, 2015, 36:1651–1662
6 Huang Y C. Unified expressions of all differential variational principles. Mech Res Commun, 2003, 30:567–572
7 Chen X W, Mei F X. Combined gradient representations of stationary nonholonomic system of Chetaev’s type(in Chinese). Chin Quart Mech,2016, 37:45–55[陈向炜,梅凤翔.定常Chetaev型非完整系统的组合梯度表示.力学季刊, 2016, 37:45–55]
8 Song D, Liu C, Guo Y X. The integral variational principles for embedded variation identity of high-order nonholonomic constrained systems(in Chinese). Acta Phys Sin, 2013, 62:094501[宋端,刘畅,郭永新.高阶非完整约束系统嵌入变分恒等式的积分变分原理.物理学报, 2013, 62:094501]
9 Haddout S. A practical application of the geometrical theory on fibered manifolds to an autonomous bicycle motion in mechanical system with nonholonomic constraints. J Geometry Phys, 2018, 123:495–506
10 Popescu P, Ida C. Nonlinear constraints in nonholonomic mechanics. J Geometric Mech, 2014, 6:527–547
11 Guo Z H, Gao P Y. On the classic nonholonomic dynamics(in Chinese). Acta Mech Sin, 1990, 22:185–190[郭仲衡,高普云.关于经典非完整力学.力学学报, 1990, 22:185–190]
12 Chen B. On contention to the classic nonholonomic dynamics(in Chinese). Acta Mech Sin, 1991, 23:379–394[陈滨.关于非完整力学的一个争议.力学学报, 1991, 23:379–394]
13 Guo Z H, Gao P Y. Further remarks on the nonholonomic dynamics(in Chinese). Acta Mech Sin, 1992, 24:253-257[郭仲衡,高普云.再关于非完整力学——答争议.力学学报, 1994, 24:253–257]
14 Alonso-Mora J, Beardsley P, Siegwart R. Cooperative collision avoidance for nonholonomic robots. IEEE Trans Robot, 2018, 34:404–420
15 Zhao Z, Liu C S, Lu J D. On nonholonomic constraints about the pure rolling of point contact(in Chinese). Acta Sci Nat Univ Pekin, 2016, 52:713 –716[赵振,刘才山,鲁建东.空间物体点接触纯滚动的几何意义.北京大学学报(自然科学版), 2016, 52:713–716]
16 Lewis A D, Murray R M. Variational principles for constrained systems:Theory and experiment. Int J Non-Linear Mech, 1995, 30:793–815
17 Mei F X. On nonholonomic mechanics(in Chinese). Mech Eng, 2015, 37:630–634[梅凤翔.关于非完整力学——分析力学札记之二十六.力学与实践, 2015, 37:630-634]
18 Barbero-Li?án M, PuiggalíM F, Ferraro S, et al. The inverse problem of the calculus of variations for discrete systems. J Phys A-Math Theor,2018, 51:185202, ar Xiv:1708.04123
19 Paraskevopoulos E, Natsiavas S. On application of Newton’s law to mechanical systems with motion constraints. Nonlinear Dyn, 2013, 72:455–475
20 Mei F X, Li Y M, Wu H B. On the Gauss principle(in Chinese). J Dyn Control, 2016, 14:301–306[梅凤翔,李彦敏,吴惠彬.关于Gauss原理.动力学与控制学报, 2016, 14:301–306]
21 Guo Y X, Luo S K, Mei F X. Progress of geometric dynamics of nonholonomic constrained mechanical systems:Lagrange theory and others(in Chinese). Adv Mech, 2004, 34:477–492[郭永新,罗绍凯,梅凤翔.非完整约束系统几何动力学研究进展:Lagrange理论及其它.力学进展,2004, 34:477–492]
22 Arnold V I, Kozlov V V, Neishtadt A I. Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia of Mathematics Science, Vol.III. Berlin:Springer, 1988
23 Fernandez O E, Bloch A M. Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data. J Phys A-Math Theor, 2008, 41:344005
24 Borisov A V, Mamaev I S, Bizyaev I A. Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry,and non-holonomic mechanics. Russ Math Surv, 2017, 72:783–840
25 Borisov A V, Kilin A A, Mamaev I S. Hamilton’s principle and the rolling motion of a symmetric ball. Dokl Phys, 2017, 62:314–317
26 Guo Y X, Zhao Z, Liu S X, et al. Conditions for Chetaev dynamics to be equivalent to vakonomic dynamics in nonholonomic systems(in Chinese). Acta Phys Sin, 2006, 55:3838–3844[郭永新,赵喆,刘世兴,等.非完整系统Chetaev动力学和vakonomic动力学的等价条件.物理学报, 2006, 55:3838–3844]
27 Llibre J, Ramírez R, Sadovskaia N. A new approach to the vakonomic mechanics. Nonlinear Dyn, 2014, 78:2219–2247
28 Mei F X. On problems of first integrals act as nonholonomic constraints(in Chinese). J Beijing Inst Tech, 1996, S1:46–50[梅凤翔.关于第一积分作为非完整约束问题.北京理工大学学报, 1996, S1:46–50]
29 Milne W E. Infinite systems of functions. Bull Amer Math Soc, 1920, 26:294–301
30 Penrose R, Todd J A. A generalized inverse for matrices. Math Proc Camb Phil Soc, 1955, 51:406–413
31 Udwadia F E, Kalaba R E. A new perspective on constrained motion. Proc R Soc A-Math Phys Eng Sci, 1992, 439:407–410
32 Chen Y H. Hamel paradox and Rosenberg conjecture in analytical dynamics. J Appl Mech, 2013, 80:041001
33 Barhorst A A. An alternative derivation of some new perspectives on constrained motion. J Appl Mech, 1995, 62:243–245
34 Foster J T. A variationally consistent approach to constrained motion. J Appl Mech, 2016, 83:054501
35 Shen H C. Routh equation of nonholonomic dynamical systems:From Chetaev condition to Euler condition(in Chinese). Acta Phys Sin, 2005,54 :2468–2473[沈惠川.一类非线性非完整系统的Routh方程:从Chetaev条件到Euler条件.物理学报, 2005, 54:2468–2473]
36 Zhao Y P. Lectures on Mechanics(in Chinese). Beijing:Science Press, 2018[赵亚溥.力学讲义.北京:科学出版社, 2018]
37 Mei F X, Wu H B, Li Y M. Two kinds of gradient representations for Nielsen equations(in Chinese). Acta Sci Nat Univ Pekin, 2016, 52:588–591[梅凤翔,吴惠彬,李彦敏. Nielsen方程的两类广义梯度表示.北京大学学报(自然科学版), 2016, 52:588–591]
2 Chen B. Analytical Dynamics(in Chinese). 2nd ed. Beijing:Peking University Press, 2012[陈滨.分析动力学.第2版.北京:北京大学出版社,2012]
3 Adamov B I, Kobrin A I. Methods of analytic mechanics in the problem of adaptive identification with constraints. Vestnik Moskov. Univ Ser 1.Mat Mekh, 2016, 5:63–67
4 Liu C S. The fundamental equations in analytical mechanics for nonholonomic systems(in Chinese). Acta Sci Nat Univ Peking, 2016, 52:756–766[刘才山.分析动力学中的基本方程与非完整约束.北京大学学报(自然科学版), 2016, 52:756–766]
5 Song H, Liang L. Investigation of power-type variational principles in liquid-filled system. Appl Math Mech-Engl Ed, 2015, 36:1651–1662
6 Huang Y C. Unified expressions of all differential variational principles. Mech Res Commun, 2003, 30:567–572
7 Chen X W, Mei F X. Combined gradient representations of stationary nonholonomic system of Chetaev’s type(in Chinese). Chin Quart Mech,2016, 37:45–55[陈向炜,梅凤翔.定常Chetaev型非完整系统的组合梯度表示.力学季刊, 2016, 37:45–55]
8 Song D, Liu C, Guo Y X. The integral variational principles for embedded variation identity of high-order nonholonomic constrained systems(in Chinese). Acta Phys Sin, 2013, 62:094501[宋端,刘畅,郭永新.高阶非完整约束系统嵌入变分恒等式的积分变分原理.物理学报, 2013, 62:094501]
9 Haddout S. A practical application of the geometrical theory on fibered manifolds to an autonomous bicycle motion in mechanical system with nonholonomic constraints. J Geometry Phys, 2018, 123:495–506
10 Popescu P, Ida C. Nonlinear constraints in nonholonomic mechanics. J Geometric Mech, 2014, 6:527–547
11 Guo Z H, Gao P Y. On the classic nonholonomic dynamics(in Chinese). Acta Mech Sin, 1990, 22:185–190[郭仲衡,高普云.关于经典非完整力学.力学学报, 1990, 22:185–190]
12 Chen B. On contention to the classic nonholonomic dynamics(in Chinese). Acta Mech Sin, 1991, 23:379–394[陈滨.关于非完整力学的一个争议.力学学报, 1991, 23:379–394]
13 Guo Z H, Gao P Y. Further remarks on the nonholonomic dynamics(in Chinese). Acta Mech Sin, 1992, 24:253-257[郭仲衡,高普云.再关于非完整力学——答争议.力学学报, 1994, 24:253–257]
14 Alonso-Mora J, Beardsley P, Siegwart R. Cooperative collision avoidance for nonholonomic robots. IEEE Trans Robot, 2018, 34:404–420
15 Zhao Z, Liu C S, Lu J D. On nonholonomic constraints about the pure rolling of point contact(in Chinese). Acta Sci Nat Univ Pekin, 2016, 52:713 –716[赵振,刘才山,鲁建东.空间物体点接触纯滚动的几何意义.北京大学学报(自然科学版), 2016, 52:713–716]
16 Lewis A D, Murray R M. Variational principles for constrained systems:Theory and experiment. Int J Non-Linear Mech, 1995, 30:793–815
17 Mei F X. On nonholonomic mechanics(in Chinese). Mech Eng, 2015, 37:630–634[梅凤翔.关于非完整力学——分析力学札记之二十六.力学与实践, 2015, 37:630-634]
18 Barbero-Li?án M, PuiggalíM F, Ferraro S, et al. The inverse problem of the calculus of variations for discrete systems. J Phys A-Math Theor,2018, 51:185202, ar Xiv:1708.04123
19 Paraskevopoulos E, Natsiavas S. On application of Newton’s law to mechanical systems with motion constraints. Nonlinear Dyn, 2013, 72:455–475
20 Mei F X, Li Y M, Wu H B. On the Gauss principle(in Chinese). J Dyn Control, 2016, 14:301–306[梅凤翔,李彦敏,吴惠彬.关于Gauss原理.动力学与控制学报, 2016, 14:301–306]
21 Guo Y X, Luo S K, Mei F X. Progress of geometric dynamics of nonholonomic constrained mechanical systems:Lagrange theory and others(in Chinese). Adv Mech, 2004, 34:477–492[郭永新,罗绍凯,梅凤翔.非完整约束系统几何动力学研究进展:Lagrange理论及其它.力学进展,2004, 34:477–492]
22 Arnold V I, Kozlov V V, Neishtadt A I. Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia of Mathematics Science, Vol.III. Berlin:Springer, 1988
23 Fernandez O E, Bloch A M. Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data. J Phys A-Math Theor, 2008, 41:344005
24 Borisov A V, Mamaev I S, Bizyaev I A. Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry,and non-holonomic mechanics. Russ Math Surv, 2017, 72:783–840
25 Borisov A V, Kilin A A, Mamaev I S. Hamilton’s principle and the rolling motion of a symmetric ball. Dokl Phys, 2017, 62:314–317
26 Guo Y X, Zhao Z, Liu S X, et al. Conditions for Chetaev dynamics to be equivalent to vakonomic dynamics in nonholonomic systems(in Chinese). Acta Phys Sin, 2006, 55:3838–3844[郭永新,赵喆,刘世兴,等.非完整系统Chetaev动力学和vakonomic动力学的等价条件.物理学报, 2006, 55:3838–3844]
27 Llibre J, Ramírez R, Sadovskaia N. A new approach to the vakonomic mechanics. Nonlinear Dyn, 2014, 78:2219–2247
28 Mei F X. On problems of first integrals act as nonholonomic constraints(in Chinese). J Beijing Inst Tech, 1996, S1:46–50[梅凤翔.关于第一积分作为非完整约束问题.北京理工大学学报, 1996, S1:46–50]
29 Milne W E. Infinite systems of functions. Bull Amer Math Soc, 1920, 26:294–301
30 Penrose R, Todd J A. A generalized inverse for matrices. Math Proc Camb Phil Soc, 1955, 51:406–413
31 Udwadia F E, Kalaba R E. A new perspective on constrained motion. Proc R Soc A-Math Phys Eng Sci, 1992, 439:407–410
32 Chen Y H. Hamel paradox and Rosenberg conjecture in analytical dynamics. J Appl Mech, 2013, 80:041001
33 Barhorst A A. An alternative derivation of some new perspectives on constrained motion. J Appl Mech, 1995, 62:243–245
34 Foster J T. A variationally consistent approach to constrained motion. J Appl Mech, 2016, 83:054501
35 Shen H C. Routh equation of nonholonomic dynamical systems:From Chetaev condition to Euler condition(in Chinese). Acta Phys Sin, 2005,54 :2468–2473[沈惠川.一类非线性非完整系统的Routh方程:从Chetaev条件到Euler条件.物理学报, 2005, 54:2468–2473]
36 Zhao Y P. Lectures on Mechanics(in Chinese). Beijing:Science Press, 2018[赵亚溥.力学讲义.北京:科学出版社, 2018]
37 Mei F X, Wu H B, Li Y M. Two kinds of gradient representations for Nielsen equations(in Chinese). Acta Sci Nat Univ Pekin, 2016, 52:588–591[梅凤翔,吴惠彬,李彦敏. Nielsen方程的两类广义梯度表示.北京大学学报(自然科学版), 2016, 52:588–591]