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 Henon 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.

中文翻译：

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等时线轨道的几何形状：从阿基米德抛物线到开普勒第三定律

在经典力学中，开普勒势和谐波势具有以下显着的特性：在这两种势中的任何一个中，束缚测试粒子以独立于角动量的径向周期运行。因此，开普勒势和谐波势称为\it{等时线}。在本文中，我们解决了以下一般问题：是否还有其他等时线势，如果有，它们包含什么样的轨道？为了回答这些问题，我们采用 Michel Henon 于 1959 年提出的几何学观点，对等时线势和等时线轨道的集合进行详尽的探索和分类。特别是，我们提供了开普勒第三定律的几何推广，并给出了任何等时线轨道的顶点角的类似定律。我们还将等时线轨道集与平面中线性变换下的抛物线集相关联，并使用它来推导出任何等时线轨道的解析参数化。在此过程中，我们将我们的结果与已知结果进行比较，指出这个数学物理问题的一些有趣细节，并认为我们的几何方法可以导出到潜在理论中更通用的轨道。