Integrable homogeneous potentials of degree -1 in the plane with small eigenvalues

2011/10/27 by Thierry Combot, Combot, Thierry
Mathematics · Physics and Astronomy · #37J30 #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #math.DS #msc:37J30 #nlin.SI

paper · pdf · doi:10.48550/arxiv.1110.6130

40 pages, 42 references

arxiv created 2015/04/18 · arxiv updated 2015/04/21

Abstract

We give a complete classification of meromorphically integrable homogeneous potentials V of degree -1 which are real analytic on ℝ²∖ \0\. In the more general case when V is only meromorphic on an open set of an algebraic variety, we give a classification of all integrable potentials having a Darboux point c with V'(c)=-c, c₁²+c₂²≠ 0 and \hboxSp(∇² V(c)) ⊂\-1,0,2\. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case V'(c)=0.

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