Existence of KAM tori in degenerate Hamiltonian systems
The work extends the standard Kolmogorov-Arnold-Moser theorem to the perturbed systems with the Hamiltonian \(H(p,q) = N(p) + P(p,q)\), where \(p\) and \(q\) are the \(n\)-dimensional vectors of the action and angle variables, and, in contrast with the usual formulation of the theorem, it is not assumed that the mapping from the action space to the frequency space, characterized by the Jacobian matrix \(\partial N/\partial p\), is nondegenerate. In this work, only real analytical Hamiltonians are considered. The demand of the nondegenerateness is replaced by an essentially weaker condition. Namely, the following theorem is proved: the small perturbation conserves almost all the invariant tori of the unperturbed system, only slightly deforming them, provided that in the domain \(\Sigma\) of analyticity of the Hamiltonian, in which the Hamiltonian is \(2 \pi\)-periodic in the angular variables \(q\), the rank \(r\) of the mapping \(\partial N/\partial p\) (\(1 \leq r \leq n\)) is constant, and the image of any open set in \(\text{Re}(\Sigma)\) includes a curved \(C^{n + 2}\) one-dimensional submanifold. The proof of the theorem is based on analysis of distribution of the Diophantine vectors in the \(n\)-dimensional Euclidean space. By foliating the action space locally into a family of curves so that an image of each curve is a curved one-dimensional manifold, the proof is restricted to the existence of KAM tori on those curves. The fact that the image manifolds do not get flat under the action of the small perturbation allows to prove that the Newtonian iteration scheme does not lose its superconvergence, provided that the Diophantine constant is changed properly at each step of the iteration.
scientific article; zbMATH DE number 696892