对H. Dullin和R. Montgomery猜想的证明
Proof of a conjecture by H. Dullin and R. Montgomery
AI总结：
本文通过开普勒极限和复分析工具，推导出新的周期公式，证明周期及其比值（旋转数）随非平凡第一积分单调变化。
摘要：
在平面欧拉问题准周期框架下，文献中已有的周期公式仅在奇点的一侧简单。本文补充了其他公式，这些公式在奇点另一侧更简洁。推导新公式的工具包括开普勒极限和复分析方法。作为应用，本文证明了H. Dullin和R. Montgomery提出的猜想：周期及其比值（旋转数）在任意固定能量水平下，都是其非平凡第一积分的单调函数。
英文摘要：
In the framework of the planar Euler problem in the quasi--periodic regime, the formulae of the periods available in the literature are simple only on one side of their singularity. In this paper, we complement such formulae with others, which result simpler on the other side. The derivation of such new formulae uses the Keplerian limit and complex analysis tools. As an application, we prove a conjecture by H. Dullin and R. Montgomery, which states that such periods, as well as their ratio, the {\it rotation number}, are monotone functions of their non--trivial first integral, at any fixed energy level.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
