A simple proof of the invariant torus theorem

We give a simple proof of Kolmogorov's theorem on the persistence of a quasiperiodic invariant torus in Hamiltonian systems. The theorem is first reduced to a well-posed inversion problem (Herman's normal form) by switching the frequency obstruction from one side of the conjugacy to another. Then the proof consists in applying a simple, well suited, inverse function theorem in the analytic category, which itself relies on the Newton algorithm and on interpolation inequalities. A comparison with other proofs is included in appendix

Lagrangian Relations and Linear Point Billiards

Motivated by the high-energy limit of the $N$-body problem we construct non-deterministic billiard process. The billiard table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed polygonal curve with vertices on the subspaces and change of direction upon hitting a subspace governed by `conservation of momentum' (mirror reflection). The itinerary of a trajectory is the list of subspaces it hits, in order. Two basic questions are: (A) Are itineraries finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In a beautiful series of papers Burago-Ferleger-Kononenko [BFK] answered (A) affirmatively by using non-smooth metric geometry ideas and the notion of a Hadamard space. We answer (B) by proving that this space of trajectories is diffeomorphic to a Lagrangian relation on the space of lines in the Euclidean space. Our methods combine those of BFK with the notion of a generating family for a Lagrangian relation.Comment: 29 pages, 4 figure

Classical nn-body scattering with long-range potentials

We consider the scattering of $n$ classical particles interacting via pair potentials, under the assumption that each pair potential is "long-range", i.e. being of order ${\cal O}(r^{-\alpha})$ for some $\alpha >0$. We define and focus on the "free region", the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an explicit, global surface of section for the free region. This surface of section is key to proving the smoothness of the map sending a point to its final state and to establishing a forward conjugacy between the $n$-body dynamics and free dynamics.Comment: 44 pages, 2 figure

About M. Herman's proof of 'Arnold's Theorem' in celestial mechanics

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La méthode des paramètres en théorie KAM

National audienc

La méthode des paramètres en théorie KAM

National audienc