Dynamical systems defining Jacobi's theta-constants

Details Dynamical systems defining Jacobi's theta-constants Dynamical systems defining Jacobi's theta-constants

2010-12-07

arxiv.org/abs/1012.1429v4

Journ. Math. Phys. (2011), 52(11), 112704(1-21)

Mathematics

Classical Analysis and ODEs

Final version. Major changes; LaTeX, 23 pages (was 17), no figures

Scientific paper

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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