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Morse theory for conditionally-periodic solutions to Hamiltonian systems. (English. Russian original) Zbl 0859.58024

Sib. Math. J. 35, No. 3, 590-604 (1994); translation from Sib. Mat. Zh. 35, No. 3, 657-673 (1994).
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).

MSC:

37C55 Periodic and quasi-periodic flows and diffeomorphisms
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34C25 Periodic solutions to ordinary differential equations
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