Curvature as an integrable deformation
arXiv:1903.09543 · doi:10.1007/978-3-030-20087-9_1
Abstract
The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the generic anisotropic oscillator Hamiltonian is presented, and its superintegrability for commensurate frequencies is shown. As a second example, an integrable version of the Hénon-Heiles system on the sphere and the hyperbolic plane is introduced. Projective Beltrami coordinates are shown to be helpful in this construction, and further applications of this approach are sketched
arXiv comment34 pages, 1 figure. To appear in "Integrability, Supersymmetry and Coherent States", A volume in honour of Professor Véronique Hussin. S. Kuru, J. Negro and L.M. Nieto (Eds.), Special volume of the CRM Series in Mathematical Physics (Berlin: Springer, 2019)
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