Meromorphically integrable homogeneous potentials with multiple Darboux points

2013/01/28 by Thierry Combot, Combot, Thierry
Mathematics · Physics and Astronomy · #37J30 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #math.DS #msc:37J30 #nlin.SI

paper · pdf · doi:10.48550/arxiv.1301.6621

22 pages

arxiv created 2013/01/28 · openalex publication_date 2013/01/28 · arxiv updated 2013/01/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We prove that the only meromorphically integrable planar homogeneous potential of degree k <> -2,0,2 having a multiple Darboux point is the potential invariant by rotation. This case is a singular case of the Maciejewski-Przybylska relation on eigenvalues at Darboux points of homogeneous potentials, and needed before a case by case special analysis. The most striking application of this Theorem is the complete classification of integrable real analytic homogeneous potentials in the plane of negative degree.

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