Dynamical systems defining Jacobi's theta-constants

TL;DR

This paper introduces a new system of differential equations that defines Jacobi's theta-constants and relates to classical dynamical systems, providing new formulations and conserved quantities.

Contribution

It presents a novel differential system for theta-constants, connecting it to classical systems and establishing Lagrangian, Hamiltonian, and Nambu formulations.

Findings

System defines theta-constants in a differentially closed way.

Establishes relationship with Jacobi's dynamical system.

Constructs nonlinear Poisson brackets and conserved quantities.

Abstract

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.