Hamiltonians with two degrees of freedom admitting a singlevalued general solution
Analysis in Theory and Applications(2005)
摘要
Following the basic principles stated by Painlevé, 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 Hénon-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 Hénon-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 explicity given by a genus two hyperelliptic function.
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关键词
two degree of freedom Hamiltonians,Painlevé test,Painlevé property,Hénon-Heiles Hamiltonian,hyperelliptic,37K15,35J60
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