Theory of tensor invariants of integrable Hamiltonian systems. I: Incompatible Poisson structures
The paper represents an interesting contribution to the theory of invariants of completely integrable Hamiltonian systems on a smooth manifold, initiated by \textit{F. Magri} [J. Math. Phys. 19, 1156-1162 (1978; Zbl 0383.35065)]. The author gives a complete classification of all invariant closed 2-forms and corresponding invariant nondegenerate Poisson structures in a neighbourhood of an invariant toroidal domain and shows that these Poisson structures are in general incompatible with the original one. Applications connected with the KAM theory, Kepler problem, celestial mechanics and the classical harmonic oscillator are presented. The concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced and related to complete integrability of a dynamical system.
scientific article; zbMATH DE number 934767
scientific article