Proceedings of the International Conference on Geometry, Integrability and Quantization

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

Angel Zhivkov and Ioanna Makaveeva

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

Source
Proceedings of the Third International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov and Gregory L. Naber, eds. (Sofia: Coral Press Scientific Publishing, 2002), 454-460

Dates
First available in Project Euclid: 12 June 2015

Permanent link to this document
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


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