# Explicit integrable systems on two dimensional manifolds with a cubic first integral - Mathematical Physics

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Abstract: A few years ago Selivanova gave an existence proof for some integrablemodels, in fact geodesic flows on two dimensional manifolds, with a cubic firstintegral. However the explicit form of these models hinged on the solution of anonlinear third order ordinary differential equation which could not beobtained. We show that an appropriate choice of coordinates allows forintegration and gives the explicit local form for the full family of integrablesystems. The relevant metrics are described by a finite number of parametersand lead to a large class of models on the manifolds \${\mb S}^2, {\mb H}^2\$ and\$P^2{\mb R}\$ containing as special cases examples due to Goryachev,Chaplygin, Dullin, Matveev and Tsiganov.

Autor: Galliano Valent LPTHE

Fuente: https://arxiv.org/