Generic classification of homogeneous potentials of degree -1 in the plane

Details Generic classification of homogeneous potentials of degree -1 in the plane Generic classification of homogeneous potentials of degree -1 in the plane

2011-10-27

arxiv.org/abs/1110.6130v1

Mathematics

Dynamical Systems

39 pages, 11 references

Scientific paper

We give a complete classification of integrable potentials in the plane V=r^{-1}U(\theta) with U meromorphic and real analytic. In the more general case V only meromorphic, we make a classification of all integrable potentials possessing a Darboux point c with V'(c)=-c and \Sp(\nabla^2 V(c)) \subset\{-1,0,2\}, and then prove the non existence of other meromorphically integrable potential such that \Sp(\nabla^2 V(c))\subset ]-\infty,20]. An algorithm is given to improve this bound arbitrary, and allows to find algorithmically all integrable potentials in any set E of potentials posseding a generic property, the eigenvalue bounded property, which corresponds to the fact that for any potential in E, the lowest eigenvalue is always lower than some fixed explicit bound. We eventually present an analysis and some examples in the degenerate Darboux point case V'(c)=0.

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