@inproceedings{060e479d8d98400fbb0d55d8d3f294d9,

title = "Painlev{\'e} property of the H{\'e}non-Heiles Hamiltonians",

abstract = "Time-independent Hamiltonians of the physical type $$ H=\frac{1}{2}(P^2_1+P^2_2)+V(Q_1,Q_2) $$ pass the Painlev{\'e} test for only seven potentials $V$, known as the H{\'e}non-Heiles Hamiltonians, each depending on a finite number of free constants. Proving the Painlev{\'e} property has not yet been 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 of such polynomial equations which possess the Painlev{\'e} property [see C. M. Cosgrove, Stud. Appl. Math. 104 (2000), no. 1, 1--65; Stud. Appl. Math. 116 (2006), no. 4, 321--413]. The properties common to each Hamiltonian are: \roster \item{"}(i){"} the general solution is meromorphic and expressed with hyperelliptic functions of genus two; \item{"}(ii){"} the Hamiltonian is complete (the addition of any time-independent term would ruin the Painlev{\'e} property).\endroster",

keywords = "Hamiltonian",

author = "Robert Conte and Micheline Musette and Caroline Verhoeven",

note = "{\'E}ric Delabaere and Mich{\`e}le Loday-Richaud; Finds and Results from the Swedish Cyprus Expedition: A Gender Perspective at the Medelhavsmuseet ; Conference date: 21-09-2009 Through 25-09-2009",

year = "2006",

language = "English",

isbn = "978-2-85629-229-7",

series = "Th{\'e}ories asymptotiques et {\'e}quations de Painlev{\'e}",

publisher = "Soci{\'e}t{\'e} Math{\'e}matique de France",

pages = "65--82",

editor = "{\'E}ric Delabaere and Mich{\`e}le Loday-richaud",

booktitle = "Seminars and Congresses",

}