Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree - 3

Jaume Llibre, Adam Mahdi, Claudia Valls

Research output: Contribution to journalArticleResearchpeer-review

5 Citations (Scopus)

Abstract

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=-2,-1,⋯,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree -3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential. © 2011 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)1928-1935
JournalPhysica D: Nonlinear Phenomena
Volume240
Issue number24
DOIs
Publication statusPublished - 1 Dec 2011

Keywords

  • Hamiltonian system with 2-degrees of freedom
  • Homogeneous potential of degree -3
  • Polynomial integrability

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