标题： 三维空间中有理Calogero-Moser模型的可积和超可积扩展 显示英文标题

标题： Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions

摘要： 我们考虑一类具有特定二次积分的3自由度哈密顿系统，该系统包括有理数Calogero-Moser系统作为特例。 对于一般类，我们引入分离坐标以找到一般的可分离（因此是Liouville可积）系统，具有两个二次积分。 这给出了Calogero-Moser系统与一大类势能的耦合，推广了在抛物线坐标中可分离的势能系列。 特例是{\em 超可积}，包括开普勒系统和共振振子。 论文的初始计算涉及平坦（笛卡尔型）动能，但在第\ref{sec:conflat-general}节，我们引入了{\em 相似因子} $\varphi$ 到$H$，并将两个二次积分扩展到这种情况。 所有先前的结果都推广到了这种情况。 然后我们引入动能的二维和三维对称代数（Killing向量），这些代数限制了共形因子。 这使我们能够将系统从3个自由度减少到2个自由度，从而产生许多有趣的系统，包括在Darboux-Koenigs$D_2$背景上的开普勒型和Hénon-Heiles型势能。 显示英文摘要

摘要： We consider a class of Hamiltonian systems in 3 degrees of freedom, with a particular type of quadratic integral and which includes the rational Calogero-Moser system as a particular case. For the general class, we introduce separation coordinates to find the general separable (and therefore Liouville integrable) system, with two quadratic integrals. This gives a coupling of the Calogero-Moser system with a large class of potentials, generalising the series of potentials which are separable in parabolic coordinates. Particular cases are {\em superintegrable}, including Kepler and a resonant oscillator. The initial calculations of the paper are concerned with the flat (Cartesian type) kinetic energy, but in Section \ref{sec:conflat-general}, we introduce a {\em conformal factor} $\varphi$ to $H$ and extend the two quadratic integrals to this case. All the previous results are generalised to this case. We then introduce some 2 and 3 dimensional symmetry algebras of the Kinetic energy (Killing vectors), which restrict the conformal factor. This enables us to reduce our systems from 3 to 2 degrees of freedom, giving rise to many interesting systems, including both Kepler type and H\'enon-Heiles type potentials on a Darboux-Koenigs $D_2$ background.