应用Udwadia-Kalaba理论对开普勒定律的研究
|
摘要
本文运用Udwadia-Kalaba理论对天体运动进行了研究(尤其针对开普勒定律和万有引力定律).我们用一种创新的方法表明天体的运行轨道可能是圆,椭圆,双曲线或抛物线.并且,在UdwadiaKalaba理论基础上,运用运行轨道约束(椭圆,圆环,双曲线或抛物线)以及角动量守恒约束核实了任何天体运动都遵从万有引力定律.基于Udwadia-Kalaba理论,我们首先考虑无约束离散动态系统,其运动方程可应用牛顿力学或拉格朗日力学以广义坐标形式写出.然后推导各类约束的二阶约束方程.最后将额外的广义力约束(从二阶约束方程获得)施加到无约束系统上.对多体系统使用此建模方法,我们总能推导出Udwadia-Kalaba方程的显式解析形式.Udwadia-Kalaba方程可用于解决完整或非完整约束问题以及理想或非理想约束问题.如果质量矩阵奇异,Udwadia-Kalaba方程也适用.
We apply the Udwadia-Kalaba theory to study the movement of heavenly bodies(especially Kepler's law and the inverse square law of gravitation). In an alternative way, we show that the orbital motion of a heavenly body may be a circle, an ellipse, a hyperbola or parabola. Also, by applying the Udwadia-Kalaba theory, we use the constraint of orbital motion(ellipse, circle, hyperbola or parabola) and the conservation of angular momentum constraint to easily verify that any heavenly body's motion complies with the inverse square law of gravitation. Using the Udwadia- Kalaba theory, we first consider the unconstrained discrete dynamic system whose equation of motion is written by applying Newtonian or Lagrangian mechanics in terms of generalized coordinates. Next, we derive the constraint equations of different kinds of constraints. Finally, we impose the additional generalized forces of constraint(obtained from the second-order constraint equations) on the unconstrained system. By using this modeling methodology for multi-body systems, we can always derive the Udwadia-Kalaba equation in an explicit analytic form. The Udwadia-Kalaba equation can be used to solve holonomic or nonholonomic constraint problems, and ideal or non-ideal constraint problems. It is also applicable to systems regardless of their mass matrices' singularity.
We apply the Udwadia-Kalaba theory to study the movement of heavenly bodies(especially Kepler's law and the inverse square law of gravitation). In an alternative way, we show that the orbital motion of a heavenly body may be a circle, an ellipse, a hyperbola or parabola. Also, by applying the Udwadia-Kalaba theory, we use the constraint of orbital motion(ellipse, circle, hyperbola or parabola) and the conservation of angular momentum constraint to easily verify that any heavenly body's motion complies with the inverse square law of gravitation. Using the Udwadia- Kalaba theory, we first consider the unconstrained discrete dynamic system whose equation of motion is written by applying Newtonian or Lagrangian mechanics in terms of generalized coordinates. Next, we derive the constraint equations of different kinds of constraints. Finally, we impose the additional generalized forces of constraint(obtained from the second-order constraint equations) on the unconstrained system. By using this modeling methodology for multi-body systems, we can always derive the Udwadia-Kalaba equation in an explicit analytic form. The Udwadia-Kalaba equation can be used to solve holonomic or nonholonomic constraint problems, and ideal or non-ideal constraint problems. It is also applicable to systems regardless of their mass matrices' singularity.
引文
1 Lagrange J L.Mechanique Analytique.Paris:Mme ve Courcier,1787
2 Gauss C F.Uber ein neues allgemeines Grundgsetz der Mechanik.J die reine und angewandte Mathematik,1829,4:232–235
3 Gibbs J W.On the fundamental formulae of dynamics.Am J Math,1879,2:49–64
4 Appell P.Sur une Forme Generale des Equations de la Dynamique.Comptes rendus de l’Academie des sciences,1899,129:459–460
5 Pars L A.A Treatise on Analytical Dynamics.Connecticut:Ox Bow Press,1965
6 Dirac P A M.Lectures in Quantum Mechanics.New York:Yeshiva University,1964
7 Udwadia F E,Kalaba R E.A new perspective on constrained motion.Proc Royal Societ,1992,439:407–410
8 Udwadia F E,Kalaba R E.Analytical Dynamics:A New Approach.Cambridg:Cambridge University Press,1996
9 Udwadia F E,Kalaba R E.Explicit equations of motion for mechanical systems with non-ideal constraints.J Appl Mech,2001,68:462–467
10 Udwadia F E,Kalaba R E.On constrained motion.Appl Math Comput,2005,164:313–320
11 Udwadia F E,Phohomsiri P.Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics.Proc Royal Soci,2006,462:2097–2117
12 Moore E H.On the reciprocal of the general algebraic matrix.Bull Am Math Soci,1920,26:294–395
13 Penrose R.A generalized inverse of matrices.Proc Cambridge Philosophical Soci,1955,51:404–413
14 Chen Y H.Second-order constraints for equations of motion of constrained systems.IEEE/ASME Trans Mech,1998,3:240–248
15 Zhao H,Zhen S C,Chen Y H.Dynamical modeling and simulation of multi-body systems by using Udwadia-Kalaba theory.Chin J Mech Eng,2013,26(5):839–850
2 Gauss C F.Uber ein neues allgemeines Grundgsetz der Mechanik.J die reine und angewandte Mathematik,1829,4:232–235
3 Gibbs J W.On the fundamental formulae of dynamics.Am J Math,1879,2:49–64
4 Appell P.Sur une Forme Generale des Equations de la Dynamique.Comptes rendus de l’Academie des sciences,1899,129:459–460
5 Pars L A.A Treatise on Analytical Dynamics.Connecticut:Ox Bow Press,1965
6 Dirac P A M.Lectures in Quantum Mechanics.New York:Yeshiva University,1964
7 Udwadia F E,Kalaba R E.A new perspective on constrained motion.Proc Royal Societ,1992,439:407–410
8 Udwadia F E,Kalaba R E.Analytical Dynamics:A New Approach.Cambridg:Cambridge University Press,1996
9 Udwadia F E,Kalaba R E.Explicit equations of motion for mechanical systems with non-ideal constraints.J Appl Mech,2001,68:462–467
10 Udwadia F E,Kalaba R E.On constrained motion.Appl Math Comput,2005,164:313–320
11 Udwadia F E,Phohomsiri P.Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics.Proc Royal Soci,2006,462:2097–2117
12 Moore E H.On the reciprocal of the general algebraic matrix.Bull Am Math Soci,1920,26:294–395
13 Penrose R.A generalized inverse of matrices.Proc Cambridge Philosophical Soci,1955,51:404–413
14 Chen Y H.Second-order constraints for equations of motion of constrained systems.IEEE/ASME Trans Mech,1998,3:240–248
15 Zhao H,Zhen S C,Chen Y H.Dynamical modeling and simulation of multi-body systems by using Udwadia-Kalaba theory.Chin J Mech Eng,2013,26(5):839–850