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| The '''Kolmogorov–Arnold–Moser theorem''' ('''KAM theorem''') is a result in ]s about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the ] of ]. | The '''Kolmogorov–Arnold–Moser theorem''' ('''KAM theorem''') is a result in ]s about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the ] that arises in the ] of ]. | ||
| The problem is whether or not a small perturbation of a ] dynamical system results in a lasting ] ]. The original breakthrough to this problem was given by ] in 1954. This was rigorously proved and extended by ] (in 1963 for analytic ]s) and ] (in 1962 for smooth ]s), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated{{clarify|date=March 2014}}, could not be applied directly as a whole to the motions of the ] because of the presence of degeneracy in the unperturbed Kepler problem. However, it is useful in generating corrections of astronomical models, and to prove long-term stability and the avoidance of ] in solar system{{why|date=March 2014}}. Arnold used the methods{{which|date=March 2014}} of KAM to prove the stability of ]s in the planar ]. | The problem is whether or not a small perturbation of a ] dynamical system results in a lasting ] ]. The original breakthrough to this problem was given by ] in 1954. This was rigorously proved and extended by ] (in 1963 for analytic ]s) and ] (in 1962 for smooth ]s), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated{{clarify|date=March 2014}}, could not be applied directly as a whole to the motions of the ] because of the presence of degeneracy in the unperturbed Kepler problem. However, it is useful in generating corrections of astronomical models, and to prove long-term stability and the avoidance of ] in solar system{{why|date=March 2014}}. Arnold used the methods{{which|date=March 2014}} of KAM to prove the stability of ]s in the planar ]. | ||
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| ===Perturbations=== | ===Perturbations=== | ||
| The KAM theorem states that if the system is subjected to a weak ], some of the invariant tori are deformed and survive, while others are destroyed.{{clarify|date=March 2014}} |
The KAM theorem states that if the system is subjected to a weak ], some of the invariant tori are deformed and survive{{clarify}}, while others are destroyed.{{clarify|date=March 2014}} Surviving tori meet the ], i.e., they have “sufficiently irrational” frequencies. This implies that the motion{{which}} continues to be ], 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 ]s, named ''Cantori'' by ] in 1979.<ref>{{Cite journal|title = A variational principle for invariant tori of fixed frequency|url = http://iopscience.iop.org/article/10.1088/0305-4470/12/3/001|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}}</ref> | ||
| ⚫ | ===Consequences=== | ||
| ⚫ | 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. | ||
| ⚫ | An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic. | ||
| ⚫ | As the perturbation increases and the smooth curves disintegrate we move from KAM theory to ] which requires less stringent hypotheses and works with the Cantor-like sets. | ||
| ⚫ | ==KAM Theory== | ||
| ⚫ | ===Consequences=== | ||
| ⚫ | 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 ]) and to systems with fast and slow frequencies (as in the work of ]). | ||
| ⚫ | An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.{{which}} | ||
| ⚫ | 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. | ||
| ⚫ | ==KAM Theory== | ||
| ⚫ | Those KAM tori that are destroyed by perturbation become invariant ]s, named ''Cantori'' by ] in 1979.<ref>{{Cite journal|title = A variational principle for invariant tori of fixed frequency|url = http://iopscience.iop.org/article/10.1088/0305-4470/12/3/001|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}}</ref> | ||
| ⚫ | 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 ]) and to systems with fast and slow frequencies (as in the work of ]). | ||
| As the perturbation increases and the smooth curves disintegrate we move from KAM theory to | |||
| ⚫ | ] which requires less stringent hypotheses and works with the Cantor-like sets. | ||
| == See also == | == See also == | ||
Revision as of 05:27, 18 January 2016
The Kolmogorov–Arnold–Moser theorem (KAM theorem) is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics.
The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Kolmogorov in 1954. This was rigorously proved and extended by Vladimir Arnold (in 1963 for analytic Hamiltonian systems) and Jürgen Moser (in 1962 for smooth twist maps), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated, could not be applied directly as a whole to the motions of the solar system because of the presence of degeneracy in the unperturbed Kepler problem. However, it is useful in generating corrections of astronomical models, and to prove long-term stability and the avoidance of orbital resonance in solar system. Arnold used the methods of KAM to prove the stability of elliptical orbits in the planar three-body problem.
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 conditions of the integrable Hamiltonian system 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, while others are destroyed. Surviving tori meet the non-resonance condition, i.e., they have “sufficiently irrational” frequencies. This implies that the motion 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 sets, 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.
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).
See also
References
- Percival, I C (1979-03-01). "A variational principle for invariant tori of fixed frequency". Journal of Physics A: Mathematical and General. 12 (3): L57 – L60. doi:10.1088/0305-4470/12/3/001.
- 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.
- Wayne, C. Eugene (January 2008). "An Introduction to KAM Theory" (PDF). Preprint: 29. Retrieved 20 June 2012.
- Jürgen Pöschel (2001). "A lecture on the classical KAM-theorem" (PDF). Proceedings of Symposia in Pure Mathematics (AMS). 69: 707–732.
- Rafael de la Llave (2001) A tutorial on KAM theory.
- Weisstein, Eric W. "Kolmogorov-Arnold-Moser Theorem". MathWorld.
- KAM theory: the legacy of Kolmogorov’s 1954 paper
- Kolmogorov-Arnold-Moser theory from Scholarpedia
- 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, ISBN 978-981-4556-58-3. Chapter 1: Introduction