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

This article is cited in 10 scientific papers (total in 10 papers)

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

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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
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\pages 888--898
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Ballesteros A., Blasco A., “Integrable Henon-Heiles Hamiltonians: A Poisson algebra approach”, Ann Physics, 325:12 (2010), 2787–2799  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    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  crossref  adsnasa  isi
    6. Lakshmanan M., Chandrasekar V.K., “Generating Finite Dimensional Integrable Nonlinear Dynamical Systems”, Eur. Phys. J.-Spec. Top., 222:3-4 (2013), 665–688  crossref  isi  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Simon S., “Conditions and Evidence For Non-Integrability in the Friedmann-Robertson-Walker Hamiltonian”, J. Nonlinear Math. Phys., 21:1 (2014), 1–16  crossref  mathscinet  adsnasa  isi  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    10. Lesfari A., “Geometric Study of a Family of Integrable Systems”, Int. Electron. J. Geom., 11:1 (2018), 78–92  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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