引力场中运动的常数与Hamilton-Jacobi函数
Constants of the Motion in a Gravitational Field and the Hamilton-Jacobi Function
摘要：
在大多数力学教科书中，首先写下牛顿定律或汉密尔顿运动方程，然后基于初始条件求解，以确定运动常数并描述粒子的轨迹。在这篇文章中，我们采取了不同的出发点。我们从广义相对论的度量开始，并展示如何通过检查运动常数来构造它们，然后可以用它们来写下轨迹方程。这将通过从度量导出一个哈密顿-雅可比函数来实现，并表明它的存在需要所有上述性质。文章最后给出了四个应用，其中包括开普勒行星运动第一定律的推导，以及描述星系在罗伯逊-沃克度规定义的空间中运动轨迹的公式。
英文摘要：
In most text books of mechanics, Newton's laws or Hamilton's equations of motion are first written down and then solved based on initial conditions to determine the constants of the motions and to describe the trajectories of the particles. In this essay, we take a different starting point. We begin with the metrics of general relativity and show how they can be used to construct by inspection constants of motion, which can then be used to write down the equations of the trajectories. This will be achieved by deriving a Hamiltonian-Jacobi function from the metric and showing that its existence requires all of the above mentioned properties. The article concludes with four applications, which includes a derivation of Kepler's First Law of Motion for planets, and a formula for describing the trajectories of galaxies moving in a space defined by the Robertson-Walker metric.
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