## Proceedings of the International Conference on Geometry, Integrability and Quantization

### Effective Solutions of an Integrable Case of the Hénon-Heiles System

#### Abstract

We solve in two-dimensional theta functions the integrable case $\ddot{r} = -ar + 2zr,\ \ddot{z} = -bz + 6z^2 + r^2$ (a and b are constant parameters) of the generalizied Hénon–Heiles system. The general solution depends on six arbitrary constants, called algebraic–geometric coordinates. Three of them are coordinates on the degree two (and dimension three) Siegel upper half-plane and define two-dimensional tori $\mathbb{T}^2.$ Each trajectory of the Hénon–Heiles system lies on certain torus $\mathbb{T}^2$. Next two arbitrary constants define the initial position on $\mathbb{T}^2$. The speed of the flow depends multiplicatively on the last arbitrary constant.

#### Article information

Dates
First available in Project Euclid: 12 June 2015

https://projecteuclid.org/ euclid.pgiq/1434145489

Digital Object Identifier
doi:10.7546/giq-3-2002-454-460

Mathematical Reviews number (MathSciNet)
MR1884867

Zentralblatt MATH identifier
1090.70012

#### Citation

Zhivkov, Angel; Makaveeva, Ioanna. Effective Solutions of an Integrable Case of the Hénon-Heiles System. Proceedings of the Third International Conference on Geometry, Integrability and Quantization, 454--460, Coral Press Scientific Publishing, Sofia, Bulgaria, 2002. doi:10.7546/giq-3-2002-454-460. https://projecteuclid.org/euclid.pgiq/1434145489