RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

 TMF, 2005, Volume 144, Number 1, Pages 14–25 (Mi tmf1827)

Completeness of the Cubic and Quartic Henon–Heiles Hamiltonians

R. Contea, M. Musetteb, C. Verhoevenb

a CEA, Service de Physique Théorique
b Vrije Universiteit

Abstract: The quartic Henon–Heiles Hamiltonian passes the Painleve test for only four sets of values of the constants. Only one of these, identical to the traveling-wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the other three have not yet been integrated in the general case $(\alpha,\beta,\gamma)\neq(0,0,0)$. We integrate them by building a birational transformation to two fourth-order first-degree equations in the Cosgrove classiffication of polynomial equations that have the Painleve property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Henon–Heiles Hamiltonians, namely, a general solution that is meromorphic and hyperelliptic with genus two in all four quartic cases. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painleve integrability (the completeness property).

Keywords: Henon–Heiles Hamiltonian, Painleve property, hyperelliptic separation of variables

DOI: https://doi.org/10.4213/tmf1827

Full text: PDF file (238 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2005, 144:1, 888–898

Bibliographic databases:

Citation: R. Conte, M. Musette, C. Verhoeven, “Completeness of the Cubic and Quartic Henon–Heiles Hamiltonians”, TMF, 144:1 (2005), 14–25; Theoret. and Math. Phys., 144:1 (2005), 888–898

Citation in format AMSBIB
\Bibitem{ConMusVer05} \by R.~Conte, M.~Musette, C.~Verhoeven \paper Completeness of the Cubic and Quartic Henon--Heiles Hamiltonians \jour TMF \yr 2005 \vol 144 \issue 1 \pages 14--25 \mathnet{http://mi.mathnet.ru/tmf1827} \crossref{https://doi.org/10.4213/tmf1827} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2194255} \zmath{https://zbmath.org/?q=an:1178.37057} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2005TMP...144..888C} \elib{http://elibrary.ru/item.asp?id=17702852} \transl \jour Theoret. and Math. Phys. \yr 2005 \vol 144 \issue 1 \pages 888--898 \crossref{https://doi.org/10.1007/s11232-005-0115-9} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000231408800003} 

• http://mi.mathnet.ru/eng/tmf1827
• https://doi.org/10.4213/tmf1827
• http://mi.mathnet.ru/eng/tmf/v144/i1/p14

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Conte R., Musette M., Verhoeven C., “Explicit integration of the Henon-Heiles Hamiltonians”, Journal of Nonlinear Mathematical Physics, 12 (2005), 212–227, Suppl. 1
2. Lesfari A., “Cyclic coverings of abelian varieties and the generalized Yang-Mills system for a field with gauge group SU(2)”, Int. J. Geom. Methods Mod. Phys., 5:6 (2008), 947–961
3. Zhao Jun-xiao, Conte R., “A connection between HH3 and Korteweg-de Vries with one source”, J. Math. Phys., 51:3 (2010), 033511, 6 pp.
4. Ballesteros A., Blasco A., “Integrable Henon-Heiles Hamiltonians: A Poisson algebra approach”, Ann Physics, 325:12 (2010), 2787–2799
5. Blasco A., Ballesteros A., Musso F., “Integrable Perturbations of Henon-Heiles Systems From Poisson Coalgebras”, XX International Fall Workshop on Geometry and Physics, AIP Conference Proceedings, 1460, eds. Linan M., Barbero F., DeDiego D., Amer Inst Physics, 2012, 159–163
6. Lakshmanan M., Chandrasekar V.K., “Generating Finite Dimensional Integrable Nonlinear Dynamical Systems”, Eur. Phys. J.-Spec. Top., 222:3-4 (2013), 665–688
7. Fre P., Sagnotti A., Sorin A.S., “Integrable Scalar Cosmologies I. Foundations and Links with String Theory”, Nucl. Phys. B, 877:3 (2013), 1028–1106
8. Simon S., “Conditions and Evidence For Non-Integrability in the Friedmann-Robertson-Walker Hamiltonian”, J. Nonlinear Math. Phys., 21:1 (2014), 1–16
9. Ballesteros A., Blasco A., Herranz F.J., Musso F., “An Integrable Henon-Heiles System on the Sphere and the Hyperbolic Plane”, Nonlinearity, 28:11 (2015), 3789–3801
10. Lesfari A., “Geometric Study of a Family of Integrable Systems”, Int. Electron. J. Geom., 11:1 (2018), 78–92
•  Number of views: This page: 337 Full text: 102 References: 63 First page: 1