Morse theory for conditionally-periodic solutions to Hamiltonian systems
The author studies perturbations of integrable Hamiltonian systems; i.e., consider the systems of differential equations \[ \dot x=\nabla_yH,\quad \dot y=-\nabla_xH\tag{1} \] with analytic Hamiltonians of the form \[ H(x,y)=H_0(y)+\varepsilon H_1(x,y),\quad (x,y)\in T^n\times D. \] Here \(x\) and \(y\) are the variables ``angle-action''. The main purpose of this article is to develop a local Morse theory for the existence problem of invariant tori of (1), which is reduced to a problem for finding critical points of a smooth function on a finite-dimensional manifold. The present method differs from the traditional method of accelerated convergence in the KAM theory, and leans on the Nash-Moser implicit function theorem, especially on the version by E. Zehnder. The author also demonstrates some applications of theorems obtained in the paper to proving existence of conditionally-periodic solutions of (1).
scientific article; zbMATH DE number 917077
scientific article