Completeness of the Cubic and Quartic Henon-Heiles Hamiltonians

Abstract

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).

Integrable Scalar Cosmologies I. Foundations and links with String Theory

“…correspond to a list of sporadic two-field dynamical systems that are Liouville integrable thanks to the existence of a conserved charge Q in addition to the Hamiltonian H. The compilation of this list was a major mathematical achievement resulting from the work of several authors (see the review papers [11,12,13] and references therein), and here we are translating these results into the cosmological setting. It is indeed remarkable that these integrable dynamical systems can be mapped into cosmological models with a potential V(ϕ) that depends only on exponentials of the scalar field ϕ.…”

“…If η is continued to purely imaginary values, these potentials correspond to the cubic and quartic generalized Hénon-Heiles integrable systems with meromorphic genus-two hyperelliptic general solutions (see [11,12,13] and references therein). Group III γ = 2 5 -(Ramani potentials) There are two scalar potentials in this class.…”

Explicit integration of the Hénon-Heiles Hamiltonians 1

“…We remark that the variable x = q 2 1 + q 2 2 obeys the fourth-order ODE 14) which, up to some translation, is identical to the ODE (5.1) in the KdV5 case.…”

A curved Hénon—Heiles system and its integrable perturbations

“…Consequently, the generalization of the Euclidean rational perturbations (12) to a constant curvature framework is achieved. In ambient coordinates (18), the potentials (27) read…”