Kolmogorov–Arnold–Moser Theorem
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The Kolmogorov–Arnold–Moser (KAM) theorem is a result in
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
. The problem is whether or not a small perturbation of a
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
dynamical system results in a lasting
quasiperiodic Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpr ...
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...
. The original breakthrough to this problem was given by
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
in 1954. This was rigorously proved and extended by
Jürgen Moser Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations. Life Moser's mother Ilse Strehl ...
in 1962 (for smooth twist maps) and
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
in 1963 (for analytic
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
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 The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
or other instances of the -body problem, but it turned out to work only for the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
because of a 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.


Statement


Integrable Hamiltonian systems

The KAM theorem is usually stated in terms of trajectories in
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
of an integrable
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
. The motion of an
integrable system In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
is confined to an invariant torus (a
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-shaped surface). Different initial conditions of the integrable
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
will trace different invariant 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 Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpr ...
, 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 In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
s, named ''Cantori'' by Ian C. Percival in 1979. 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.


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) and to systems with fast and slow frequencies (as in the work of Mikhail B. Sevryuk).


KAM torus

A manifold \mathcal^ invariant under the action of a flow \phi^ is called an invariant d-torus, if there exists a diffeomorphism \boldsymbol:\mathcal^\rightarrow \mathbb^ into the standard d-torus \mathbb^:=\underbrace _ such that the resulting motion on \mathbb^ is uniform linear but not static, ''i.e.'' \mathrm\boldsymbol / \mathrmt = \boldsymbol ,where \boldsymbol\in\mathbb^ is a non-zero constant vector, called the ''frequency vector''. If the frequency vector \boldsymbol is: * rationally independent (''a.k.a.'' incommensurable, that is \boldsymbol\cdot\boldsymbol \neq 0 for all \boldsymbol\in\mathbb^\backslash\left\) * and "badly" approximated by rationals, typically in a ''Diophantine'' sense: \exist~ \gamma, \tau > 0 \text , \boldsymbol\cdot\boldsymbol, \geq \frac, \forall ~\boldsymbol\in\mathbb^\backslash \left\ , then the invariant d-torus \mathcal^ (d\geq 2) is called a ''KAM torus''. The d=1 case is normally excluded in classical KAM theory because it does not involve small divisors.


See also

*
Stability of the Solar System The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ...
*
Arnold diffusion In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusi ...
*
Ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
*
Hofstadter's butterfly In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovere ...
*
Nekhoroshev estimates The Nekhoroshev estimates are an important result in the theory of Hamiltonian systems concerning the long-time stability of solutions of integrable systems under a small perturbation of the Hamiltonian. The first paper on the subject was written ...


Notes


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. * * * Rafael de la Llave (2001)
A tutorial on KAM theory
'. *
KAM theory: the legacy of Kolmogorov’s 1954 paper

Kolmogorov-Arnold-Moser theory
from
Scholarpedia ''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine. ''Scholarpedia'' articles are written ...
* H Scott Dumas.
The KAM Story – A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory
', 2014, World Scientific Publishing, .
Chapter 1: Introduction
' {{DEFAULTSORT:Kolmogorov-Arnold-Moser theorem Hamiltonian mechanics Theorems in dynamical systems Computer-assisted proofs