应用Udwadia-Kalaba理论对开普勒定律的研究

On Kepler's law:Application of the Udwadia-Kalaba theory

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.