On the Darboux integrability of the logarithmic galactic potentials

Jaume Llibre, Clàudia Valls

Research output: Contribution to journalArticleResearchpeer-review


© 2017 Elsevier B.V. We study the logarithmic Hamiltonians H=(px2+py2)∕2+log(1+x2+y2∕q2)1∕2, which appear in the study of the galactic dynamics. We characterize all the invariant algebraic hypersurfaces and all exponential factors of the Hamiltonian system with Hamiltonian H. We prove that this Hamiltonian system is completely integrable with Darboux first integrals if and only if q=±1.
Original languageEnglish
Pages (from-to)279-287
JournalJournal of Geometry and Physics
Publication statusPublished - 1 Nov 2017


  • Darboux first integrals
  • Darboux polynomials
  • Invariant algebraic hypersurfaces
  • Logarithmic galactic potential
  • Polynomial integrability
  • Rational integrability


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