An integrable Hénon–Heiles system on the sphere and the hyperbolic plane

Abstract

We construct a constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the integrable Hénon–Heiles Hamiltonian given byH=12( p12+p22)+Ω(q12+4q22)+α(q12q2+2q23), ?>where and α are real constants. The curved integrable Hamiltonian so obtained depends on a parameter which is just the curvature of the underlying space, and is such that the Euclidean Hénon–Heiles system is smoothly obtained in the zero-curvature limit . On the other hand, the Hamiltonian that we propose can be regarded as an integrable perturbation of a known curved integrable anisotropic oscillator. We stress that in order to obtain the curved Hénon–Heiles Hamiltonian , the preservation of the full integrability structure of the flat Hamiltonian 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 of integrable polynomial potentials, in which the flat Hénon–Heiles potential can be embedded, will be essential in our construction. Such an infinite family of curved RDG potentials on and will also be explicitly presented.