Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2006-04-06
Nonlinear Sciences
Exactly Solvable and Integrable Systems
20 pages, 0 figure, Angers 1-5 June 2004. Se'minaires et congre`s, Socie'te' mathe'matique de France, to appear
Scientific paper
Time independent Hamiltonians of the physical type H = (P_1^2+P_2^2)/2+V(Q_1,Q_2) pass the Painleve' test for only seven potentials $V$, known as the He'non-Heiles Hamiltonians, each depending on a finite number of free constants. Proving the Painleve' property was not yet achieved for generic values of the free constants. We integrate each missing case by building a birational transformation to some fourth order first degree ordinary differential equation in the classification (Cosgrove, 2000) of such polynomial equations which possess the Painleve' property. The properties common to each Hamiltonian are: (i) the general solution is meromorphic and expressed with hyperelliptic functions of genus two, (ii) the Hamiltonian is complete (the addition of any time independent term would ruin the Painleve' property).
Conte Robert
Musette Micheline
Verhoeven Caroline
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