论文标题
投射的可整合机械台球
Projective Integrable Mechanical Billiards
论文作者
论文摘要
在本文中，我们将投影性动力学方法用于整合机械台球，如[Zhao，2021]中，建立了自然机械台球与拉格朗日问题的整合性，这是两个开普勒问题的叠加和两个霍克问题的叠加，而Hooke的彼此与基于基础的机械中心的Hooke中心，以及与基础的组合组合在一起，并将其组合在一起。作为反射壁，在平面，球体和双曲平面中。这涵盖了许多以前已知的综合机械台球，尤其是飞机上的可集成的胡克，开普勒和两个中心的台球，正如[Takeuchi-Zhao，2021]中所研究的，作为子案例。 [Takeuchi-Zhao，2021]基于共形对应关系的方法也已应用于双曲机平面中的可集成的开普勒台球，以说明它们与半球和双曲机上相应的可相应的Hooke台球的等效性。
In this paper, we use the projective dynamical approach to integrable mechanical billiards as in [Zhao, 2021] to establish the integrability of natural mechanical billiards with the Lagrange problem, which is the superposition of two Kepler problems and a Hooke problem, with the Hooke center at the middle of the Kepler centers, as the underlying mechanical systems, and with any combinations of confocal conic sections with foci at the Kepler centers as the reflection wall, in the plane, on the sphere, and in the hyperbolic plane. This covers many previously known integrable mechanical billiards, especially the integrable Hooke, Kepler and two-center billiards in the plane, as has been investigated in [Takeuchi-Zhao, 2021], as subcases. The approach of [Takeuchi-Zhao, 2021] based on conformal correspondence has been also applied to integrable Kepler billiards in the hyperbolic plane to illustrate their equivalence with the corresponding integrable Hooke billiards on the hemisphere and in the hyperbolic plane as well.
