Systems of Hess-Appel'rot type and Zhukovskii property

Abstract

We start with a review of a class of systems with invariant relations, so called systems of Hess-Appel'rot type that generalizes the classical Hess-Appel'rot rigid body case. The systems of Hess-Appel'rot type have remarkable property: there exists a pair of compatible Poisson structures, such that a system is certain Hamiltonian perturbation of an integrable bi-Hamiltonian system. The invariant relations are Casimir functions of the second structure. The systems of Hess-Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the Zhukovskii property: These are Hamiltonian systems on a symplectic manifold M with actions of two groups G and K; the systems are assumed to be K-invariant and to have invariant relation Φ = 0 given by the momentum mapping of the G-action, admitting two types of reductions, a reduction to the Poisson manifold P = M/K and a partial reduction to the symplectic manifold N0 = Φ-1(0)/G; final and crucial assumption is that the partially reduced system to N0 is completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess-Appel'rot type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess-Appel'rot type the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian Gr+(n, 2) has a natural interpretation within Zhukovskii property and that it is equivalent to a partial reduction of certain system of HessAppel'rot type. We perform a classical and algebro-geometric integration of the system in dimension four, as an example of a known isoholomorphic - system the Lagrange bitop. The paper presents a lot of examples of systems of Hess-Appel'rot type, giving an additional argument in favor of further study of this class of systems.