By Dmitry Treschev

This booklet offers the elemental equipment of standard perturbation concept of Hamiltonian platforms, together with KAM-theory, splitting of asymptotic manifolds, the separatrix map, averaging, anti-integrable restrict, and so forth. in a readable approach. even if concise, it discusses all major points of the elemental smooth conception of perturbed Hamiltonian structures and such a lot effects are given with entire proofs.

It may be a priceless reference for Hamiltonian platforms, and of precise curiosity to researchers and graduate scholars of the KAM community.

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Then, for a generic potential V , there exists a trajectory with p(t) → ∞ as t → ∞. In an unpublished manuscript [89] Mather obtained the following result. Let M be a two-dimensional torus. Then there exists a minimizing closed geodesic σ in an arbitrary simple homotopy class Γ of closed curves in M. If the minimizing geodesic in Γ is unique, it follows from the results of Morse [91] that σ 50 2 Introduction to the KAM Theory possesses a homoclinic13 trajectory γ . 14 There exist a, b ∈ R such that dist(γ (t), σ (t + a)) → 0 as t → −∞, dist(γ (t), σ (t + b)) → 0 as t → ∞.

Nonintegrability of the perturbed system prevents analyticity of Ψ in all arguments. 4. The condition of nondegeneracy can be weakened considerably if we look not at an individual torus, but at the whole family of nonresonant tori. In particular, the following theorem was anounced by Rüssmann (the proof is given in [121, 122]). Let the system be analytic and suppose that the image of the map y → ν(y) does not belong to any hyperplane in Rm ν . Then for small ε the perturbed system has invariant tori.

N = 0. s,u are Lagrangian. 2. For any w ∈ N the subspaces Tw N ⊕ Ew u × E s : if v ∈ E u and 3. The form ω defines a nondegenerate bilinear form on Ew u w w s ω(vu , vs ) = 0 for all vs ∈ Ew , then vu = 0. Proof. The first statement follows from the Herman theorem [59]. The argument is as follows. 14)) on Tn is preserved by the differential equation x˙ = ν. Let us put ω0 = αj k (x) dxj ∧ dxk . 5 Hyperbolic Tori: Weak Reducibility 41 The vector field ν, ∂/∂x is a restriction of the Hamiltonian vector field vH to N .

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