Editing Kolmogorov–Arnold–Moser theorem (section)

{{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>