等时轨道的几何学：从阿基米德抛物线到开普勒第三定律
The Geometry of Isochrone Orbits: from Archimedes' parabolae to Kepler's third law
摘要：
在经典力学中，开普勒势和调和势都有一个显著的特性：在这两种势中，一个受束缚的测试粒子的轨道的径向周期与它的角动量无关。因此，开普勒势和调和势被称为\it{isochrone}。在本文中，我们解决了以下一般问题：是否存在其他的等时势，如果有的话，它们包含什么样的轨道？为了回答这些问题，我们采用了Michel Hénon于1959年提出的几何观点，对等时势和等时轨道集进行了详尽的探索和分类。特别是，我们提供了一个几何推广开普勒第三定律，并给出了类似的法律的拱点角，任何等时线轨道。我们也将等时线轨道的集合与线性变换下的平面中的抛物线的集合相关联，并使用它来导出任何等时线轨道的解析参数化。一路上，我们将我们的结果与已知的结果进行比较，指出这个数学物理问题的一些有趣的细节，并认为我们的几何方法可以导出到更一般的轨道在潜在的理论。
英文摘要：
In classical mechanics, the Kepler potential and the Harmonic potential share the following remarkable property: in either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and Harmonic potentials are called \it{isochrone}. In this paper, we solve the following general problem: are there any other isochrone potentials, and if so, what kind of orbits do they contain? To answer these questions, we adopt a geometrical point of view initiated by Michel Hénon in 1959, in order to explore and classify exhaustively the set of isochrone potentials and isochrone orbits. In particular, we provide a geometric generalization of Kepler's third law, and give a similar law for the apsidal angle, for any isochrone orbit. We also relate the set of isochrone orbits to the set of parabolae in the plane under linear transformations, and use this to derive an analytical parameterization of any isochrone orbit. Along the way we compare our results to known ones, pinpoint some interesting details of this mathematical physics problem, and argue that our geometrical methods can be exported to more generic orbits in potential theory.
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