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Dynamical systems defining Jacobi's ϑ-constants

### 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.

© 2011 American Institute of Physics

Received 24 July 2011
Accepted 28 October 2011
Published online 29 November 2011

Acknowledgments:
The research was supported by the Federal Targeted Program under state contracts 02.740.11.0238, #P1337 and #P22. The work by SLL and AASH was supported by the RFBR grant 09–02–00723-a and SLL had a partial support from the RFBR grant 08–01–00737-a. SLL and AASH appreciate the hospitality of the Erwin Schroedinger Institute for Mathematical Physics, Vienna.

Article outline:

I. INTRODUCTION
A. Motivation for the work
B. The paper content
II. ODES DEFINING ϑ-CONSTANTS
A. On symmetrical system (8)
B. An integrable modification of system (8)
III. EXPLICIT SOLUTIONS AND TECHNICALITIES
A. Associated linear ODEs
B. Solution to the Jacobi system
C. Solution to system (19)
IV. DERIVATION OF INTEGRALS
V. INTEGRALS AND LAGRANGIAN
A. Conserved quantities
1. On the Ramamani system
B. Action and Lagrangians
VI. POISSON STRUCTURES
VII. A GENERALIZATION