A hierarchy of multidimensional Hénon-Heiles systems
The original Hénon-Heiles system is described by a classical Hamiltonian with two degrees of freedom. The potential is a polynom of third order containing two parameters. The special choice of these parameters makes the system integrable in the Liouville sense. Its equations of motions were interpreted by \textit{A. P. Fordy} as stationary flows of some soliton equations of fifth order [Physica D 52, No. 2/3, 204--210 (1991; Zbl 0753.35102)]. The author constructs and studies a hierarchy of multidimensional generalizations of Hénon-Heiles systems obtained as constrained flows of the Korteweg-de Vries hierarchy. The last method, formulated by \textit{M. Antonowicz} and \textit{S. Rauch-Wojciechowski} [Phys. Lett. A 147, No. 8-9, 455--462 (1990)], generates finite-dimensional integrable systems from a given infinite-dimensional integrable system (constrained (or restricted) flows are essentially a new parameterization of stationary flows). The author derives (in rather technical way) the Lax representation, classical Poisson structure and \(r\)-matrix for the obtained hierarchy. Finally, he separates variables in the Hamilton-Jacobi equation (defining new variables as zeros of some polynom associated with the Lax representation) and constructs a Jacobi inversion problem for the hierarchy.
scientific article; zbMATH DE number 1530096
scientific article