Explicit integration of the Hénon-Heiles Hamiltonians 1

Conte, Robert; Musette, Micheline; Verhoeven, Caroline

doi:10.2991/jnmp.2005.12.s1.18

Cited by 34 publications

(27 citation statements)

References 41 publications

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“…It converts the local information into the global one and can be used not only as an alternative to the standard method, but also as an addition to it, which assists to find solutions of the obtained algebraic system. We have demonstrated that one can find elliptic solutions of the generalized Hénon-Heiles system solving only linear equations and nonlinear equations in one variable, instead of nonlinear system (11). At the same time, to use this method one has to know not only an algebraic system, but also the differential equations from which this system has been obtained.…”

Section: Resultsmentioning

confidence: 99%

“…The general solutions of the Hénon-Heiles system are known only in integrable cases [37,38,11], in other cases not only four-, but even three-parameter exact solutions have yet to be found. The generalized Hénon-Heiles system has attracted enormous attention over the years and has used as a model in astronomy [30] and in gravitation [25,32].…”

Section: The Hénon-heiles Systemmentioning

confidence: 99%

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Construction of Special Solutions for Nonintegrable Systems

Vernov¹

2006

JNMP

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The Painlevé test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and a search for solutions of the obtained system. It has been demonstrated by the example of the generalized Hénon-Heiles system that the use of the Laurent series solutions of the initial differential equation assists to solve the obtained algebraic system. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial differential equation in the form of the Laurent or Puiseux series we use the Painlevé test. This test can also assist to solve the inverse problem: to find the form of a polynomial potential, which corresponds to the required type of solutions. We consider the five-dimensional gravitational model with a scalar field to demonstrate this.

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Section: Resultsmentioning

confidence: 99%

Section: The Hénon-heiles Systemmentioning

confidence: 99%

Construction of Special Solutions for Nonintegrable Systems

Vernov¹

2006

JNMP

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“…The integrability and non-integrability related to the Hénon-Heiles problem with some parameters and with D = 0 has been considered by many authors, as for example, [8], [9], [13], [15], [17], [18], [22], [23], [28].…”

Section: Introductionmentioning

confidence: 99%

Periodic Solutions, Stability and Non-Integrability in a Generalized Hénon-Heiles Hamiltonian System

Carrasco-Olivera¹,

Vidal²

2021

JNMP

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We consider the Hamiltonian function defined by the cubic polynomial H = 1 2 (p 2 x + p 2 y ) + 1 2 (x 2 + y 2 ) + A 3 x 3 + Bxy 2 + Dx 2 y, where A, B, D ∈ R are parameters and so H is an extension of the well known Hénon-Heiles problem. Our main contribution for D = 0, A + B = 0 and other technical restrictions are in three aspects: existence of periodic solutions, stability and instability of these periodic solutions and the problem of nonintegrability of the system associated to H. Initially we give sufficient conditions on the three parameters of these generalized Hénon-Heiles systems, which guarantees that at any positive energy level, the Hamiltonian system has periodic orbits. After that, we prove that its stability changes with the values of the parameters. Finally, we show that the generalized Hénon-Heiles systems cannot have any second first integral of class C 1 in the sense of Liouville-Arnol'd. In fact, the parameters where our problem is not integrable in the sense of Liouville-Arnol'd are the same where the periodic orbits were analytically found through averaging theory.

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“…3, we recall all the cases (three "cubic" and four "quartic") where the most general classical timeindependent Hamiltonian with two degrees of freedom can have a single-valued general solution. Discarding the integrated cases (see [2] for a review of the current state of this problem), we then focus on the three cases (all "quartic") where the general solution is still missing, aiming to find their general solution.…”

Section: Introductionmentioning

confidence: 99%

Completeness of the Cubic and Quartic Henon-Heiles Hamiltonians

2005

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The quartic Hénon-Heiles Hamiltonian passes the Painlevé 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 (α, β, γ) = (0, 0, 0). We integrate them by building a birational transformation to two fourth-order firstdegree equations in the Cosgrove classification of polynomial equations that have the Painlevé property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Hénon-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 Painlevé integrability (the completeness property).

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Explicit integration of the Hénon-Heiles Hamiltonians 1

Cited by 34 publications

References 41 publications

Construction of Special Solutions for Nonintegrable Systems

Construction of Special Solutions for Nonintegrable Systems

Periodic Solutions, Stability and Non-Integrability in a Generalized Hénon-Heiles Hamiltonian System

Completeness of the Cubic and Quartic Henon-Heiles Hamiltonians

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