Abstract
Separable Hamiltonian systems either in sphero-conical coordinates on an S 2 sphere or in elliptic coordinates on a \({\mathbb R}^2\) plane are described in a unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with a spherical configuration space to its Liouville Type I partners where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context.
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Acknowledgements
The authors thank the Spanish Ministerio de Economía y Competitividad (MINECO) for financial support under grant MTM2014-57129-C2-1-P and the Junta de Castilla y León (Projects VA057U16, VA137G18, and BU229P18). We gratefully acknowledge the constructive comments on the paper offered by the anonymous referee.
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León, M.A. ., Guilarte, J.M., Mayado, M.d.l.T. (2019). On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere. In: Kuru, Ş., Negro, J., Nieto, L. (eds) Integrability, Supersymmetry and Coherent States. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-20087-9_16
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