Painlevé property of the Hénon-Heiles Hamiltonians

Robert Conte, Micheline Musette, Caroline Verhoeven

Onderzoeksoutput: Conference paper


Time-independent Hamiltonians of the physical type $$ H=\frac{1}{2}(P^2_1+P^2_2)+V(Q_1,Q_2) $$ pass the Painlevé test for only seven potentials $V$, known as the Hénon-Heiles Hamiltonians, each depending on a finite number of free constants. Proving the Painlevé 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 Painlevé 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 Painlevé property).\endroster
Originele taal-2English
TitelSeminars and Congresses
RedacteurenÉric Delabaere, Michèle Loday-richaud
UitgeverijSociété Mathématique de France
Aantal pagina's13
ISBN van geprinte versie978-2-85629-229-7
StatusPublished - 2006
EvenementUnknown - Stockholm, Sweden
Duur: 21 sep 200925 sep 2009

Publicatie series

NaamThéories asymptotiques et équations de Painlevé



Bibliografische nota

Éric Delabaere and Michèle Loday-Richaud


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