A proof of the invariant torus theorem of Kolmogorov
英文摘要:
Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.
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Jacques Féjoz
Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Kolmogorov invariant torus theorem for weakly interacting particles I: Full dimensional tori
本文提出一个抽象KAM定理,用于研究无限多相互作用粒子系统,构建全维KAM tori以应对长程相互作用。
We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which a…
In this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the function $Ψ$ of two variables defined on semi-cylinder is established. It is proved that if unperturbed Hamiltonian has a saddle point, then under arbitrary perturbations there persists at least one hyperbolic torus.
In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighborhood of a given $ I^{0}$ for nearly integrable and analytic Hamiltonian system $ H(I,θ) = H_{0}(I)+ εH_{1}(I,θ)$ of infinite degree of freedom and of short range. That is to say, we will give an extension of the original Kolmogorov theorem to the infinite-dimensional case of short range. The proof is based on the approximation of finite-dimensional Kolmogorov theorem and an improved KAM machinery which works for the normal form depending on initial $ I^{0}$.
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.
Kolmogorov-Arnold-Moser Theorem
本文沿 Kolmogorov 在 ICM 报告中的原始思路详细给出 KAM 定理的严格证明,以澄清其论证能否严谨化,并区别于 Arnold 后来采用不同方案发表的证明。
An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers
本文基于Bourgain的迪菲antine型非共振频率或更弱条件,证明了无限维哈密顿系统中全维KAM环面的持久性,并首次证明了Aubry-MacKay猜想中频率保持的呼吸子存在性。
We show that the geometry-of-numbers method used by A. Bounemoura to obtain filling times for linear flow on the torus satisfying Diophantine conditions may be extended to the case of linear flow with truncated Diophantine conditions, and we use these methods to recover the optimal estimate first obtained by M. Berti, L. Biasco, and P. Bolle in 2003. We also briefly review the dynamics of linear flow on the torus, previous results, optimality, and applications of these estimates.
A proof that all the sliding trajectories of generic inelastic piecewise linear dynamical systems over the torus are closed
本文研究环面上的3维分段光滑非弹性线性向量场,分类了切点集并描述了滑动轨迹行为,证明了一般条件下其所有滑动轨迹均闭合,并给出了拓扑等价性结果。
This is part I of a book on KAM theory. We start from basic symplectic geometry, review Darboux-Weinstein theorems action angle coordinates and their global obstructions. Then we explain the content of Kolmogorov's invariant torus theorem and make it more general allowing discussion of arbitrary invariant Lagrangian varieties over general Poisson algebras. We include it into the general problem of normal forms and group actions. We explain the iteration method used by Kolmogorov by giving a finite dimensional analog. Part I explains in which context we apply the theory of Kolmogorov spaces whi…
We give a short proof that the ergodic sums of $\mathcal{C}^1$ observables for a $\mathcal{C}^1$ flow on $\mathbb{T}^2$ admitting a closed transversal curve whose Poincaré map has constant type rotation number have growth deviating at most logarithmically from a linear one. For this, we relate the latter integral to the Birkhoff sum of a well-chosen observable on the circle and use the Denjoy-Koksma inequality. We also give an example of a nonminimal flow satisfying the above assumptions.
In the paper, we prove an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible systems. Using this KAM theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of reversible (non-Hamiltonian) coupled nonlinear Schrödinger systems on $d-$torus $\mathbb{T}^d$.
In 1967 Moser proved the existence of a normal form for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. In this paper we present a proof of existence of this normal form based on an abstract inverse function theorem in analytic class. The given geometrization of the proof can be opportunely adapted accordingly to the specificity of systems under study. In this more conceptual frame, it becomes natural to show the existence of new remarkable normal forms, and provide several translated-torus theorems or twisted-torus theorems for s…
We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial con…
We show that an analytic invariant torus $\cT_0$ with Diophantine frequency $ø_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\cT_0$ satisfies a Rüssmann transversality condition, the torus $\cT_0$ is accumulated by KAM tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency $ø_0$. For frequency vectors $ø_0$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), …
I defended my PhD Thesis in Rome, Università Roma Tre, on April, 23, 2009, under the direction of Professor Luigi Chierchia. The judging committee was composed by Professors M. Berti, A. Celletti, C. Falcolini, J. Féjoz. Professors M. Berti and J. Féjoz refereed my thesis. The main result of my thesis is the first direct proof (the first general proof was given in [J. Féjoz, ETDS, 2004]) of a famous statement by V. I. Arnold (1963), usually referred to as "Arnold's Planetary Theorem". My proof of Arnold's Planetary Theorem relies on the rediscovery, during the year 2008, of a symplectic set of…
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