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KAM theory in configuration space. (English) Zbl 0682.58014
A Hamiltonian system is considered in the 2n-dimensional space and the Hamiltonian is supposed to be periodic in the first n coordinates, i.e. the system is defined on $$T^ n\times R^ n$$ where $$T^ n$$ is the n- dimensional torus. The problem is transformed into a variational problem whose Euler-Lagrange equation gives rise to a nonlinear partial differential equation. The latter’s solution is a diffeomorphism on $$T^ n$$. This solution gives rise to invariant tori of prescribed periods. The proofs are intricate. The existence proof is based upon a Newton type iteration technique. For given prescribed periods the uniqueness of the corresponding invariant torus is also proved.
Reviewer: M.Farkas

MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37C80 Symmetries, equivariant dynamical systems (MSC2010)
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