Persistence Of Lower-Dimensional Invariant Tori In Generalized Hamiltonian Systems

The classical KAM theory which is constructed by three famous mathematicians Kolmogorov, Arnold and Moser in the last century is the landmark of the development of Hamiltonian system. It made the stability of solar system got reasonable explanation and brought a new method into the study of Hamiltonian systems. The classical KAM theory which constructed on 2n-dimensional smoothly manifold asserted that under the Kolmogorov non-degenerate condition, the majority of the non-resonant tori of integrable system are persistent under small perturbation. After several ten years, the KAM theory has been developed into a more completer theory. First of all, the smoothness demand is weakened. Secondly, the classical KAM theory is generalized to some kinds of degenerate circumstance(the Hessian matrix of the unperturbed system is degenerate), for example Bruno([10]), Russmann([57,58]), Cheng and Sun([16]), Xu, You and Qiu([70]), Sevryuk([63]), etc. They proved the persistent of KAM invariant tori under some partial non-degenerate conditions. Among all the non-degenerate conditions, the weakest condition seems to be the Rvissmann geometry condition: the frequency does not lie in any hyperplane which pass through the origin. In paper [70], Xu, You and Qiu proved that under the analytic condition, Russmann geometry condition is equiva-lent to the following:The classical KAM theory is also generalized to symplectic manifold n,l + n = even. The symplectic forms above are assumed to have constant coefficients. The case of non-constant coefficients was treated by Cong and Li([21]).For the classical Hamiltonian systems on symplectic manifold, the symplectic structure brings some special properties. Since there is no symplectic structure for odd-dimensional system, some results in classical Hamiltonian systems are no longer hold, the development of the KAM theory for odd-dimensional system has been considered as a challenging problem([22,44,64]). The KAM type of theory-has been developed for volume preserving flows in the work of Broer, Huitema, and Takens([8]), Broer, Huitema and Sevryuk([9]). For diff'eomorphisms which are either volume preserving or satisfying the intersection property was treated in the work of Cheng and Sun([8]), Xia([69]), Cong, Li and Huang([20]).The persistence of lower-dimemsional invariant tori under small perturbation is another study direction of the KAM theory. On one hand, it provided a effective method for the study of invariant tori on the resonant surface, for example Treshchev([67]), Cong, Kiipper, Li and You([19]), Li and Yi([36]). On the other hand, the study of the persistence of lower dimensional invariant tori under small perturbation is also important on its own right. In 1967, Melnikov announced that under some conditions(the second Melnikov condition), the majority of elleptic-type lower dimensional invariant tori will persist under a small perturbation, the complete proof was carried out fifteen years later by Eliasson([24]), Kuksin([35]), Poschel([54]). Under the first Melnikov condition, similar result was given by Bour-gain([ll]), Xu and You([71]). For the persistence of hyperbolic-type lower dimensional invariant tori was treated in the work of Chierchia and Gallavotti([17]), Elias-son([25]), Graff([29]), Rudnev and Wiggins([56]), Treshchev([67]), Zehnder([75]).In the spirit of the KAM theory of generalized Hamitonian systems and thepersistence of lower dimensional invariant tori for the classical Hamiltonian systems, we concern in this paper the persistence of the lower dimensional invariant tori for generalized Hamiltonian systems.Consider the Poisson manifold , where G Rl is a bounded, connected, closed region, Tn is the standard n-torus, l, n, m are positive intergers. The structure matrix which associated with 2-form w2is a real analytic, anti-symmetric, matrix valued function and satisfies the following two conditions: (i)rank I > 0; (ii)Jacobi identityfor all The object we study is required to be invariant relative to Tn. The unperturbed system is...