Hamiltonians with two degrees of freedom admitting a singlevalued general solution

Details Hamiltonians with two degrees of freedom admitting a singlevalued general solution Hamiltonians with two degrees of freedom admitting a singlevalued general solution

2005-07-07

arxiv.org/abs/nlin/0507012v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

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

Scientific paper

Following the basic principles stated by Painlev\'e, 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 H\'enon-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 H\'enon-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.

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