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Curvature as an Integrable Deformation

  • Ángel Ballesteros
  • Alfonso Blasco
  • Francisco J. HerranzEmail author
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

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.

Keywords

Integrable systems Curvature Sphere Hyperbolic plane Integrable perturbations Oscillator potential Hénon–Heiles 

Notes

Acknowledgements

This work has been partially supported by Ministerio de Ciencia, Innovación y Universidades (Spain) under grant MTM2016-79639-P (AEI/FEDER, UE) and by Junta de Castilla y León (Spain) under grant BU229P18.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ángel Ballesteros
    • 1
  • Alfonso Blasco
    • 2
  • Francisco J. Herranz
    • 2
    Email author
  1. 1.Departamento de Física, Facultad de CienciasUniversidad de BurgosBurgosSpain
  2. 2.Departamento de Física, Escuela Politécnica SuperiorUniversidad de BurgosBurgosSpain

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