Editing Kolmogorov–Arnold–Moser theorem

{{Short description|Result in dynamical systems}}
The '''Kolmogorov–Arnold–Moser''' ('''KAM''') '''theorem''' is a result in [[dynamical system]]s about the persistence of [[quasiperiodic motion]]s under small perturbations.  The theorem partly resolves the [[small-divisor problem]] that arises in the [[perturbation theory]] of [[mechanics|classical mechanics]].

The problem is whether or not a small perturbation of a [[conservative force|conservative]] dynamical system results in a lasting [[Quasiperiodic motion|quasiperiodic]] [[orbit (dynamics)|orbit]]. The original breakthrough to this problem was given by [[Andrey Kolmogorov]] in 1954.<ref>A. N. Kolmogorov, "On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian [О сохранении условнопериодических движений при малом изменении функции Гамильтона]," ''Dokl. Akad. Nauk SSR'' '''98''' (1954).</ref> This was rigorously proved and extended by [[Jürgen Moser]] in 1962<ref>J. Moser, "On invariant curves of area-preserving mappings of an annulus," ''Nachr. Akad. Wiss.'' Göttingen Math.-Phys. Kl. II '''1962''' (1962), 1–20.</ref> (for smooth [[twist map]]s) and [[Vladimir Arnold]] in 1963<ref>V. I. Arnold, "Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian [Малые знаменатели и проблема устойчивости движения в классической и небесной механике]," ''Uspekhi Mat. Nauk'' '''18''' (1963) (English transl.: ''Russ. Math. Surv.'' '''18''', 9--36, doi:10.1070/RM1963v018n05ABEH004130 ).</ref> (for analytic [[Hamiltonian system]]s), and the general result is known as the KAM theorem.

Arnold originally thought that this theorem could apply to the motions of the [[Solar System]] or other instances of the [[n-body problem|{{mvar|n}}-body problem]], but it turned out to work only for the [[three-body problem]] because of a [[Degeneracy (mathematics)|degeneracy]] in his formulation of the problem for larger numbers of bodies. Later, [[Gabriella Pinzari]] showed how to eliminate this degeneracy by developing a rotation-invariant version of the theorem.<ref>{{citation|last=Khesin|first=Boris|author-link=Boris Khesin|editor-last=Colliander|editor-first=James|editor-link=James Colliander|title=Addendum to Arnold Memorial Workshop: Khesin on Pinzari's talk|work=James Colliander's Blog|date=October 24, 2011|url=http://blog.math.toronto.edu/colliand/2011/10/24/addendum-to-arnold-memorial-workshop-khesin-on-pinzaris-talk/|access-date=March 29, 2017|archive-url=https://web.archive.org/web/20170329142909/http://blog.math.toronto.edu/colliand/2011/10/24/addendum-to-arnold-memorial-workshop-khesin-on-pinzaris-talk/|archive-date=March 29, 2017|url-status=dead}}</ref>

==Statement==

===Integrable Hamiltonian systems===

The KAM theorem is usually stated in terms of trajectories in [[phase space]] of an integrable [[Hamiltonian system]]. The motion of an [[integrable system]] is confined to an [[invariant torus]] (a [[doughnut]]-shaped surface).  Different [[initial condition]]s of the integrable Hamiltonian system will trace different invariant [[torus|tori]] in phase space.  Plotting the coordinates of an integrable system would show that they are quasiperiodic.

===Perturbations===

The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, i.e. there is a map from the original manifold to the deformed one that is continuous in the perturbation. Conversely, other invariant tori are destroyed: even arbitrarily small perturbations cause the manifold to no longer be invariant and there exists no such map to nearby manifolds.  Surviving tori meet the non-resonance condition, i.e., they have “sufficiently irrational” frequencies. This implies that the motion on the deformed torus continues to be [[quasiperiodic]], with the independent periods changed (as a consequence of the non-degeneracy condition). The KAM theorem quantifies the level of perturbation that can be applied for this to be true.

Those KAM tori that are destroyed by perturbation become invariant [[Cantor set]]s, named ''Cantori'' by [[Ian C. Percival]] in 1979.<ref>{{Cite journal|title = A variational principle for invariant tori of fixed frequency|journal = Journal of Physics A: Mathematical and General|date = 1979-03-01|pages = L57–L60|volume = 12|issue = 3|doi = 10.1088/0305-4470/12/3/001|first = I C|last = Percival|bibcode = 1979JPhA...12L..57P }}</ref>

The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom.  As the number of dimensions of the system increases, the volume occupied by the tori decreases.

As the perturbation increases and the smooth curves disintegrate we move from KAM theory to Aubry–Mather theory which requires less stringent hypotheses and works with the Cantor-like sets.

The existence of a KAM theorem for perturbations of quantum many-body integrable systems is still an open question, although it is believed that arbitrarily small perturbations will destroy integrability in the infinite size limit.

===Consequences===

An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.{{which|date=January 2016}}

==KAM theory==

The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasiperiodic motions, now known as '''KAM theory'''.  Notably, it has been extended to non-Hamiltonian systems (starting with Moser), to non-perturbative situations (as in the work of [[Michael Herman (mathematician)|Michael Herman]]) and to systems with fast and slow frequencies (as in the work of Mikhail B. Sevryuk).

== KAM torus ==
A manifold <math>\mathcal{T}^{d}</math> invariant under the action of a flow <math>\phi^{t}</math> is called an invariant <math>d</math>-torus, if there exists a diffeomorphism <math>\boldsymbol{\varphi}:\mathcal{T}^{d}\rightarrow \mathbb{T}^{d}</math>  into the standard <math>d</math>-torus <math>\mathbb{T}^{d}:=\underbrace{ \mathbb{S}^{1}\times\mathbb{S}^{1}\times\cdots\times\mathbb{S}^{1}}
_{d}</math>  such that the resulting motion on <math>\mathbb{T}^{d}</math> is uniform linear but not static, ''i.e.'' <math>\mathrm{d}\boldsymbol{\varphi} / \mathrm{d}t = \boldsymbol{\omega}
</math>，where <math>\boldsymbol{\omega}\in\mathbb{R}^{d}</math> is a non-zero constant vector, called the ''frequency vector''. 

If the frequency vector <math>\boldsymbol{\omega}</math> is: 

* rationally independent (''a.k.a.'' incommensurable, that is <math>\boldsymbol{k}\cdot\boldsymbol{\omega} \neq 0</math> for all <math>\boldsymbol{k}\in\mathbb{Z}^{d}\backslash\left\{ \boldsymbol{0} \right\}</math>)
* and "badly" approximated by rationals, typically in a ''Diophantine'' sense:  <math>\exist~ \gamma, \tau > 0 \text{ such that }
|\boldsymbol{\omega}\cdot\boldsymbol{k}|\geq \frac{\gamma}{\|\boldsymbol{k}\|^{\tau}},
\forall ~\boldsymbol{k}\in\mathbb{Z}^{d}\backslash \left\{\boldsymbol{0} \right\}
</math>,

then the invariant <math>d</math>-torus <math>\mathcal{T}^{d}</math> (<math>d\geq 2</math>) is called a ''KAM torus''. The <math>d=1</math> case is normally excluded in classical KAM theory because it does not involve small divisors.

== See also ==
*[[Stability of the Solar System]]
*[[Arnold diffusion]]
*[[Ergodic theory]]
*[[Hofstadter's butterfly]]
*[[Nekhoroshev estimates]]

== Notes ==
{{Reflist}}

== References ==
* Arnold, Weinstein, Vogtmann. ''Mathematical Methods of Classical Mechanics'', 2nd ed., Appendix 8: Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem. Springer 1997.
* Sevryuk, M.B. ''Translation of the V. I. Arnold paper “From Superpositions to KAM Theory” (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740). Regul. Chaot. Dyn. 19, 734–744 (2014)''. https://doi.org/10.1134/S1560354714060100
* {{cite journal|last=Wayne|first=C. Eugene|title=An Introduction to KAM Theory|journal=Preprint|date=January 2008|pages=29|url=http://math.bu.edu/people/cew/preprints/introkam.pdf|access-date=20 June 2012}}
* {{cite journal
 | author = Jürgen Pöschel
 | title = A lecture on the classical KAM-theorem
 | journal = Proceedings of Symposia in Pure Mathematics
 | volume = 69
 | year = 2001
 | pages = 707–732
 | url = http://www.poschel.de/pbl/kam-1.pdf
 | doi = 10.1090/pspum/069/1858551
 | citeseerx = 10.1.1.248.8987
 | isbn = 9780821826829
 | access-date = 2006-06-06
 | archive-date = 2016-03-03
 | archive-url = https://web.archive.org/web/20160303194559/http://www.poschel.de/pbl/kam-1.pdf
 | url-status = dead
 }}
* Rafael de la Llave (2001) ''[http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=01-29 A tutorial on KAM theory]''.
* {{mathworld |urlname=Kolmogorov-Arnold-MoserTheorem|title=Kolmogorov-Arnold-Moser Theorem}}
* [http://www.math.rug.nl/~broer/pdf/kolmo100.pdf KAM theory: the legacy of Kolmogorov’s 1954 paper]
* [http://www.scholarpedia.org/article/KAM_theory_in_celestial_mechanics Kolmogorov-Arnold-Moser theory] from [[Scholarpedia]]
* H Scott Dumas. ''[http://www.worldscientific.com/worldscibooks/10.1142/8955 The KAM Story – A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory]'', 2014, World Scientific Publishing, {{ISBN|978-981-4556-58-3}}. ''[http://www.worldscientific.com/doi/suppl/10.1142/8955/suppl_file/8955_chap01.pdf Chapter 1: Introduction]''

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[[Category:Hamiltonian mechanics]]
[[Category:Theorems in dynamical systems]]
[[Category:Computer-assisted proofs]]