An integrable Henon-Heiles system on the sphere and the hyperbolic plane

Abstract

We construct a constant curvature analogue on the two-dimensional sphere ${\mathbf{S}}^2$ and the hyperbolic space ${\mathbf{H}}^2$ of the integrable Hénon-Heiles Hamiltonian $\mathcal{H}$ given by $$\mathcal{H}=\frac{1}{2}(p_1^2+p_2^2)+\Omega \left(q_1^2+4q_2^2\right)+\alpha \left(q_1^2{q}_2+2q_2^3\right),$$ where $\Omega$ and $\alpha$ are real constants. The curved integrable Hamiltonian ${\mathcal{H}}_{\kappa }$ so obtained depends on a parameter $\kappa$ which is just the curvature of the underlying space, and is such that the Euclidean Hénon-Heiles system $\mathcal{H}$ is smoothly obtained in the zero-curvature limit $\kappa \to 0$. On the other hand, the Hamiltonian ${\mathcal{H}}_{\kappa }$ that we propose can be regarded as an integrable perturbation of a known curved integrable $1:2$ anisotropic oscillator. We stress that in order to obtain the curved Hénon-Heiles Hamiltonian ${\mathcal{H}}_{\kappa }$, the preservation of the full integrability structure of the flat Hamiltonian $\mathcal{H}$ under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani-Dorizzi-Grammaticos (RDG) series ${\mathcal{V}}_n$ of integrable polynomial potentials, in which the flat Hénon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved RDG potentials ${\mathcal{V}}_{\kappa ,n}$ on ${\mathbf{S}}^2$ and ${\mathbf{H}}^2$ will be also explicitly presented.

14 pages. Comments and references added. To appear in Nonlinearity

References in corpus (8)

• Reduction of superintegrable systems: the anisotropic harmonic oscillator
• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• Relative equilibria in the 3-dimensional curved n-body problem
• Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
• The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane
• A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane
• An integrable hierarchy with a perturbed Henon-Heiles system
• Integrable Henon-Heiles Hamiltonians: a Poisson algebra approach

Cited by in corpus (2)

• A curved Henon-Heiles system and its integrable perturbations
• Curvature as an integrable deformation