Integrable and superintegrable extensions of the rational Calogero-Moser model in 3 dimensions
Integrable and superintegrable extensions of the rational Calogero-Moser model in 3 dimensions
复制标题
有理 Calogero-Moser 模型在 3 维中的可积和超可积扩展
DOI：
10.1088/1751-8121/ac6403
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Qing Huang
中科院分区：
文献类型：
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作者：
Allan P. Fordy;Qing Huang
抽象的。我们考虑一类3自由度的Hamilton系统，具有一种特殊类型的二次积分，其中包括有理Calogero-Moser系统作为一个特殊情况。对于一般类，我们引入分离坐标来找到一般的可分（因此刘维尔可积）系统，有两个二次积分。这给出了一个耦合的Calogero-Moser系统与一大类的潜力，概括了一系列的潜力是可分离的抛物线坐标。特殊情况下是超可积的，包括开普勒和共振振荡器。本文的初始计算涉及平坦（笛卡尔型）动能，但在第5节中，我们对H引入一个保角因子φ，并将两个二次积分推广到这种情况。所有以前的结果推广到这种情况。然后，我们引入了一些2维和3维的动能（Killing向量）的对称代数，它限制了共形因子。这使我们能够减少我们的系统从3到2自由度，从而产生了许多有趣的系统，包括开普勒型和Hénon-Heiles型的潜力达布-柯尼希斯D2背景。
Abstract. 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 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 5, we introduce a conformal factor φ 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 Hénon-Heiles type potentials on a Darboux-Koenigs D2 background.