力学中的类比Raychaudhuri方程
Analog Raychaudhuri equation in mechanics
摘要：
通常，在力学中，我们通过在选定的初始条件下求解牛顿第二定律来获得粒子在给定力场中的轨迹。相比之下，通过我们的工作在这里，我们首先展示了如何可以分析一个适当定义的家庭的轨迹的一个给定的机械系统的行为。这种方法使我们发展出一种力学模拟，它遵循著名的Raychaudhuri方程，该方程在黎曼几何和广义相对论中得到了广泛的研究。相对论者更熟悉的测地线聚焦的概念，似乎类似于机械系统在有限时间内的轨迹相遇。应用我们的一般结果的情况下，简单的摆，我们得到有关的定量结果。此后，我们建立和执行一个简单的实验的基础上，系统与两个摆。实验结果与我们提出的理论模型相吻合。总之，简单的理论以及相关的实验为我们提供了一种从基本观点理解高等物理学中一个相当复杂的概念的本质的方法。
英文摘要：
Usually, in mechanics, we obtain the trajectory of a particle in a given force field by solving Newton's second law with chosen initial conditions. In contrast, through our work here, we first demonstrate how one may analyse the behaviour of a suitably defined family of trajectories of a given mechanical system. Such an approach leads us to develop a mechanics analog following the well-known Raychaudhuri equation largely studied in Riemannian geometry and general relativity. The idea of geodesic focusing, which is more familiar to a relativist, appears to be analogous to the meeting of trajectories of a mechanical system within a finite time. Applying our general results to the case of simple pendula, we obtain relevant quantitative consequences. Thereafter, we set up and perform a straightforward experiment based on a system with two pendula. The experimental results on this system are found to tally well with our proposed theoretical model. In summary, the simple theory, as well as the related experiment, provides us with a way to understand the essence of a fairly involved concept in advanced physics from an elementary standpoint.
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