paper

Hamiltonians with two degrees of freedom admitting a singlevalued general solution

arXiv:nlin/0507012 · doi:10.1007/BF02836923

Abstract

Following the basic principles stated by PainlevÃ©, we first revisit the process of selecting the admissible time-independent Hamiltonians $H = (p_{1}^{2} + p_{2}^{2}) /2 + V (q_{1}, q_{2})$ whose some integer power $q_{j}^{n_{j}} (t)$ of the general solution is a singlevalued function of the complex time $t$ . In addition to the well known rational potentials $V$ of HÃ©non-Heiles, this selects possible cases with a trigonometric dependence of $V$ on $q_{j}$ . Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven (three ``cubic'' plus four ``quartic'') rational HÃ©non-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.

10 pages, Nanjing, 20-24 July 2004. To appear, Analysis in theory and applications (Nanjing)