Dynamical systems and ergodic theory Hamiltonian dynamics Theorem

Kolmogorov–Arnold–Moser theorem

Builds on Hamiltonian system Torus Integrable system Action-angle coordinates

Statement

Let $n \ge 2$ , let $D \subseteq \R^n$ be a bounded open set and let $\mathbb{T}^n = \R^n / 2\pi \Z^n$ . Use action-angle coordinates $(I, \theta)$ on $D \times \mathbb{T}^n$ , with symplectic form $\sum_k dI_k \wedge d\theta_k$ . Consider the Hamiltonian system with Hamiltonian $H_\varepsilon(I, \theta) = H_0(I) + \varepsilon H_1(I, \theta),$ where $H_0$ and $H_1$ are real-analytic on a neighbourhood of $\overline{D} \times \mathbb{T}^n$ . Assume that $H_0$ is nondegenerate: $\det \left( \partial^2 H_0 / \partial I^2 \right) \ne 0$ on $\overline{D}$ . Then there is $\varepsilon_0 > 0$ such that for $0 \le \varepsilon < \varepsilon_0$ :

the phase space contains a family of invariant $n$ -tori of $H_\varepsilon$ . Each is close to an invariant torus $\set{I = I_0}$ of $H_0$ with Diophantine frequency vector $\omega = \nabla H_0(I_0)$ . This means that $\abs{\langle k, \omega \rangle} \ge \gamma \abs{k}^{-\tau}$ for all $k \in \Z^n \setminus \set{0}$ , for some constants $\gamma > 0$ and $\tau > n - 1$ ;
on each of these tori the flow of $H_\varepsilon$ is conjugate to the linear flow $\theta \mapsto \theta + t \omega$ ;
the Lebesgue measure of the complement of the union of these tori in $D \times \mathbb{T}^n$ tends to $0$ as $\varepsilon \to 0$ .

Remarks

The proof is a Newton-type iteration. At each step a canonical change of coordinates removes the angle dependence of the perturbation to a higher order. Each step needs to divide by $\langle k, \omega \rangle$ , the small divisors, and the Diophantine condition controls them. The quadratic convergence of the Newton scheme beats the loss caused by the small divisors.

Kolmogorov announced the result in 1954, Arnold proved it for analytic Hamiltonians, and Moser proved a version for twist maps of finite smoothness.

Tori with resonant frequencies, where $\langle k, \omega \rangle = 0$ for some $k \ne 0$ , are in general destroyed by the perturbation. Some nondegeneracy condition on $H_0$ is needed, because the proof uses the map $I \mapsto \nabla H_0(I)$ to choose tori with Diophantine frequencies; other versions use the isoenergetic nondegeneracy condition instead.

Going deeper

For a student

KAM theory answers a question of Poincaré on the persistence of quasi-periodic motion in nearly integrable systems. Its key ingredients are the Diophantine condition on frequencies and a rapidly converging iteration. For two degrees of freedom the surviving $2$ -tori separate each $3$ -dimensional energy level, which gives stability of the actions for all time. In three or more degrees of freedom the tori do not separate the energy levels, and Arnold diffusion becomes possible; the Nekhoroshev estimates bound its speed. The same ideas apply to area-preserving twist maps such as the standard map.

Reading path

To understand this page from the ground up, read these 67 pages in order. The foundations come first.

Action-angle coordinates concept
Element definition
Hamiltonian system concept
Integrable system concept
Proposition concept
Real numbers object
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Complement definition
Function definition
Basis of a topology definition
Subspace topology definition
Equivalence relation definition
Product topology definition
Preimage of a set definition
Metric space definition
Codomain definition
Function composition definition
Injective function definition
Equivalence class definition
Open ball definition
Surjective function definition
Metric topology definition
Quotient set definition
Bijective function definition
Open set definition
Quotient topology definition
Inverse function definition
Hausdorff space definition
Open cover definition
Continuous map of topological spaces definition
Closed set definition
Compact space definition
Complement of an open set is closed statement
Homeomorphism definition
Universal property of the quotient topology statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example
Torus example
Kolmogorov–Arnold–Moser theorem statement

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Tag	`3ZZSXKM16Z`
URI	`m2://3ZZSXKM16Z`
Kind	statement, theorem
Writing	W0 stub
MSC	37J40, 70H08
Status	This copy of the site has no build data. Run `m2 status 3ZZSXKM16Z`.
About	Integrable system
Uses	Hamiltonian system, Torus, Integrable system, Action-angle coordinates
Same as	Wikidata Q587081 Wikipedia Kolmogorov–Arnold–Moser theorem

Kolmogorov–Arnold–Moser theorem

Statement

Remarks

Going deeper

For a student

Reading path

Where this leads

Further reading

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