用历史方法讲授开普勒第二定律
A Historical Method Approach to Teaching Kepler's 2nd law
摘要：
开普勒第二定律，面积定律，通常在第一定律和第三定律之间被顺便教授，“稍后”作为角动量守恒的结果被解释。第一定律和第三定律受到了大量的关注;第一定律是因为它在彻底改变托勒密和哥白尼的周转圆模型中的范式转变意义，第三定律是因为它方便了标准课程，因为它有一个简单的数学陈述，允许定量作业和考试。在这项工作中，我提出了一种方法，用于教学的第二定律，结合了第一和第三的数学倾向范式转变的意义。这种方法植根于历史方法，事实上，放在历史背景中，开普勒的第二定律和第一定律一样具有革命性：正如第一定律废除了震中，第二定律废除了对点。这种教授第二定律的方式也定量地表述了“时间=面积”的陈述，以开普勒方程的方式，M = E-esin E（涉及平均异常M，偏心异常E和偏心率e），其中左手边是时间，右手边是面积。在这样做的过程中，它自然地为完成模块的主动学习计算练习铺平了道路，例如，计算火星下一次冲日的时间和位置。这种方法部分基于开普勒的原始思想，因此最好适用于研究型学生，如大三和大四物理/天文学本科生或研究生。
英文摘要：
Kepler's 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained "later on" as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the 1st law because of the paradigm shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler's 2nd is as revolutionary as the 1st: as the 1st law does away with the epicycle, the 2nd law does away with the equant. This way of teaching the 2nd law also formulates the "time=area" statement quantitatively, in the way of Kepler's equation, M = E - e sin E (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars' next opposition. This method is partially based on Kepler's original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.
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