On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere

  • Miguel A.  González LeónEmail author
  • Juan Mateos Guilarte
  • Marina de la Torre Mayado
Part of the CRM Series in Mathematical Physics book series (CRM)


Separable Hamiltonian systems either in sphero-conical coordinates on an S2 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.


Separation of variables Sphero-conical coordinates Elliptic coordinates Liouville dynamical systems Trajectory isomorphism 



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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Miguel A.  González León
    • 1
    Email author
  • Juan Mateos Guilarte
    • 2
  • Marina de la Torre Mayado
    • 2
  1. 1.Departamento de Matemática Aplicada and IUFFyMUniversidad de SalamancaSalamancaSpain
  2. 2.Departamento de Física Fundamental and IUFFyMUniversidad de SalamancaSalamancaSpain

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