定义Jacobi θ常数的动力系统
Dynamical systems defining Jacobi's theta-constants
摘要：
我们提出了一个方程组，定义维尔斯特拉斯-雅可比的η-和θ-常数系列在一个差分封闭的方式。这个系统被证明有一个鲜为人知的动力系统雅可比获得的直接关系。经典上已知的微分方程达布-赫伦，查齐，和拉马努金是微分后果或减少这些系统。所提出的系统被证明承认拉格朗日，哈密顿，和Nambu配方。我们明确地构造了一束非线性泊松括号和完整的对合守恒量。作为该理论的副产品，我们还研究了Ramamani动力系统和二次系统的守恒量.
英文摘要：
We propose a system of equations that defines Weierstrass--Jacobi's eta- and theta-constant series in a differentially closed way. This system is shown to have a direct relationship to a little-known dynamical system obtained by Jacobi. The classically known differential equations by Darboux--Halphen, Chazy, and Ramanujan are the differential consequences or reductions of these systems. The proposed system is shown to admit the Lagrangian, Hamiltonian, and Nambu formulations. We explicitly construct a pencil of nonlinear Poisson brackets and complete set of involutive conserved quantities. As byproducts of the theory, we exemplify conserved quantities for the Ramamani dynamical system and quadratic system of Halphen--Brioschi.
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