Bi-Hamiltonian structure of an integrable Henon-Heiles system

Quasi-bi-Hamiltonian systems and separability

“…On the other hand, in the case of finite-dimensional systems arising as restricted or stationary flows from soliton equations [3,4], the final result of the reduction procedure are some physically interesting dynamical systems (for example the Hénon-Heiles system) which, in their natural phase space, satisfy a weaker condition than the biHamiltonian one. So, the notion of quasi-biHamiltonian (QBH) system can be introduced [5,6]; it was applied in [7] to dynamical systems with two degrees of freedom. One of the aims of this paper is just to give explicit examples of QBH systems with more than two degrees of freedom.…”

On the integrability of stationary and restricted flows of the KdV hierarchy

“…For this reason, we propose an integrability criterion holding for a generic finite-dimensional Hamiltonian system. It generalizes the criterion introduced in [8] for the particular case of the Hénon-Heiles system. Though weaker than the bi-Hamiltonian scheme, it assures Liouville-integrability of a Hamiltonian system [9] in its standard phase space, i.e.…”

“…5.2. To this purpose, let us make the following choices: i) Q 1 = E, the vector field Z 1 := X 1 (2.41) with Hamiltonian h 0 := H 0 (2.36); ii) the tensor field N := N H (5.2) and Q 0 := P −2 = N This integrability structure is related, through the map (2.38), to the one introduced in [8] for the Hamiltonian (2.37) with a 4 = 0.…”

The quasi-bi-Hamiltonian formulation of the Lagrange top

“…The qbH model was introduced in [15,2] and developed in [3,16] (see also [4] and references therein). Here we summarise some facts to be used in the rest of the paper.…”