Separability and Lax pairs for Hénon–Heiles system
- 1 June 1993
- journal article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 34 (6) , 2385-2393
- https://doi.org/10.1063/1.530123
Abstract
The Hamiltonian system corresponding to the (generalized) Hénon–Heiles Hamiltonian H= 1/2(px2+py2)+1/2Ax2+1/2By2+x2y+εy3 is known to be integrable in the following three cases: (A=B, ε=1/3); (ε=2); (B=16A, ε=16/3). In the first two the system has been integrated by making use of genus one and genus two theta functions. We show that in the third case the system can also be integrated by making use of elliptic functions. Finally, using the Fairbanks theorem, we find Lax pairs for each of the three integrable systems under consideration.Keywords
This publication has 12 references indexed in Scilit:
- The Hénon-Heiles system revisitedPhysica D: Nonlinear Phenomena, 1991
- Nonintegrability of Hénon-Heiles system and a theorem of ZiglinKodai Mathematical Journal, 1985
- The complete Whittaker theorem for two-dimensional integrable systems and its applicationJournal of Physics A: General Physics, 1983
- A theory of exact and approximate configurational invariantsPhysica D: Nonlinear Phenomena, 1983
- Branching of solutions and the nonexistence of first integrals in Hamiltonian mechanics. IIFunctional Analysis and Its Applications, 1983
- Painleve property and integrals of motion for the Henon-Heiles systemPhysics Letters A, 1982
- Analytic structure of the Henon–Heiles Hamiltonian in integrable and nonintegrable regimesJournal of Mathematical Physics, 1982
- A connection between nonlinear evolution equations and ordinary differential equations of P-type. IJournal of Mathematical Physics, 1980
- On the Stability of Isolating Integrals. I. Effect of the Perturbation in the Potential FunctionJournal of the Physics Society Japan, 1972
- The applicability of the third integral of motion: Some numerical experimentsThe Astronomical Journal, 1964