论文标题
等距轨道的几何形状：从阿基米德的抛物线到开普勒的第三定律
The Geometry of Isochrone Orbits: from Archimedes' parabolae to Kepler's third law
论文作者
论文摘要
在经典力学中，开普勒电势和谐波电位具有以下出色特性：在这些电位中的任何一个中，具有径向周期的结合测试粒子轨道与其角动量无关。因此，开普勒和谐波电势称为\ it {等句}。在本文中，我们解决了以下一般问题：是否还有其他同弦电位，如果是，它们包含什么样的轨道？为了回答这些问题，我们采用了米歇尔·海农（MichelHénon）于1959年发起的几何观点，以详尽地探索和分类等质电位和等速异隆轨道。特别是，我们提供了开普勒第三定律的几何概括，并为任何等质轨道提供了类似的apsidal角度定律。我们还将一组等异隆轨道与线性转换下平面的抛物线组合相关联，并使用它来得出任何等体轨道的分析参数化。在途中，我们将结果与已知结果进行比较，查明了此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潜在理论中更通用的轨道。
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.
