Abstract

We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero–Moser system. At the discrete level, the Lax pairs with a parameter are introduced and the discrete-time equations of motion are obtained as together with the corresponding discrete-time Lagrangian. The integrability property of this new system can be expressed in terms of the discrete Lagrangian closure relation by using a connection with the temporal Lax matrices of the discrete-time Ruijsenaars–Schneider system, an exact solution, and the existence of a classical \(r\)-matrix. As the parameter tends to zero, the standard Calogero–Moser system is recovered in both discrete-time and continuous-time forms.

中文翻译：

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单参数离散时间 Calogero–Moser 系统

摘要

我们提出了一种新型可积一维多体系统，称为单参数 Calogero-Moser 系统。在离散层面，引入带参数的Lax对，得到离散时间运动方程以及相应的离散时间拉格朗日量。这个新系统的可积性质可以用离散拉格朗日闭包关系来表达，通过使用与离散时间 Ruijsenaars-Schneider 系统的时间 Lax 矩阵的连接、精确解以及经典\(r \） -矩阵。当参数趋于零时，标准Calogero-Moser系统以离散时间和连续时间形式恢复。