Nonlinear Dynamics

, Volume 93, Issue 2, pp 933–943 | Cite as

On the integrability of 2D Hamiltonian systems with variable Gaussian curvature

  • A. A. Elmandouh
Original Paper


In this work, we consider the integrability of a general 2D motion of a particle on a surface with variable Gaussian curvature under the influence of conservative potential forces. Although this system has a kinetic energy relying on the coordinates, it remains homogeneous. The homogeneity of the system generally enables us to find a particular solution that can be utilized to derive the necessary conditions for the integrability by studying the properties of the differential Galois group of the normal variational equations along this particular solution. We present a new theory that can be applied to determine the necessary conditions for the integrability of Hamiltonian systems having a variable Gaussian curvature.


Liouville integrability Differential Galois theory Systems in polar coordinates 



We would like to express our thanks to the reviewers for their useful comments which enabled us to improve the presentation of this article.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceKing Faisal UniversityAl-AhsaaSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

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