A curved Hénon—Heiles system and its integrable perturbations

30th International Colloquium on Group Theoretical Methods in Physics

Ballesteros, Angel^1 ; Blasco, Alfonso^1 ; Herranz, Francisco J.^1

Departamento de Flsica, Universidad de Burgos, E-09001 Burgos, Spain^1

关键词: Constant curvature;    Euclidean;    Henon-heiles systems;    Homogeneous polynomials;    Hyperbolic spaces;    Integrability;    Negative index;    Real constants;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012013/pdf
DOI  :  10.1088/1742-6596/597/1/012013

来源: IOP

【 摘 要 】

The constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the integrable Henon-Heiles Hamiltonian H given by where Ω and are real constants, is revisited. The resulting integrable curved Hamiltonian, Hκkappa;, depends on a parameter κ which is just the curvature of the underlying space and allows one to recover H under the smooth flat/Euclidean limit κ → 0. This system can be regarded as an integrable cubic perturbation of a specific curved 1 : 2 anisotropic oscillator, which was already known in the literature. The Ramani series of potentials associated to Hκkappa; is fully constructed, and corresponds to the curved integrable analogues of homogeneous polynomial perturbations of H that are separable in parabolic coordinates. Integrable perturbations of Hκkappa; are also fully presented, and they can be regarded as the curved counterpart of integrable rational perturbations of the Euclidean Hamiltonian H. It will be explicitly shown that the latter perturbations can be understood as the 'negative index' counterpart of the curved Ramani series of potentials. Furthermore, it is shown that the integrability of the curved Henon-Heiles Hamiltonian Hκkappa; is preserved under the simultaneous addition of curved analogues of 'positive' and 'negative' families of Ramani potentials.