论文标题

教授开普勒第二定律的历史方法

A Historical Method Approach to Teaching Kepler's 2nd law

论文作者

Lyra, Wladimir

论文摘要

开普勒的第二定律,即该地区的法律,通常在第一和第三法之间传授,这是由于角度保护的结果,以“以后”解释。第一和第三法律受到了很多关注;第一定律是因为范式转移的意义在用托勒密和哥白尼的表周进行大修时,这是第三定律,这是因为它在具有简单的数学陈述方面具有简单的数学陈述,从而可以进行定量的家庭作业和考试。在这项工作中,我推进了一种教授第二定律的方法,该方法结合了第一定律的范式偏移意义和第三定律的数学倾向。这种方法源于历史方法,实际上是在其历史背景下,开普勒的第二次与第一法一样革命性:随着第一定律消除了epicycle,第二定律消除了等价。这种教授第二定律的方式还定量地制定了“时间=区域”陈述,以开普勒方程式,m = e-e sin e(相关的平均异常M,偏心异常E和偏心率E),左侧是时间,右侧是右侧的区域。这样一来,它自然而然地铺平了通过主动学习计算练习来完成模块的方法,例如计算火星下一个对立的时间和位置。该方法部分基于开普勒的原始思想,因此应最好地应用于以研究为导向的学生,例如初级和高级物理/天文学本科生或研究生。

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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