Kolmogorov–Arnold–Moser theorem

The Kolmogorov -Arnold -Moser theorem (in short " KAM theorem ") is a result from the theory of dynamical systems, which makes statements about the behavior of such a system under small perturbations. The theorem solves partially the problem of small divider that in the perturbation theory of dynamical systems, especially in celestial mechanics, emerges.

The KAM theorem arose from the question of whether a small perturbation of a conservative dynamical system leads to a quasi-periodic motion. The breakthrough in answering this question succeeded Kolmogorov in 1954. The result was rigorously proved in 1963 by Moser for so-called smooth twist maps and 1964 by Arnold for Hamiltonian systems.

Heuristics

The main result of the KAM theory guarantees the existence of quasi-periodic solutions for a certain class of differential equations. An important subclass of which form the differential equations for the so-called N- body problem. Quasi Periodic solutions can be close together, but between them can be unstable orbits, so that in practice can not be decided because, for example, finite measurement accuracy whether you are on a stable or unstable path. For the planetary system can be shown that the unstable orbits are much rarer than the stable.

The theorem

If an undisturbed system is not degenerate, then for sufficiently small autonomous Hamiltonian disorders most non-resonant tori only slightly deformed, so that the phase space of the perturbed system invariant tori exist that are wound tightly and quasi-periodic of the phase paths, the frequencies are rationally independent. These invariant tori form a majority in the sense that the measure of the complement of their union is small when the disorder is weak.

Credentials

  • Jürgen Pöschel: A lecture on the classical KAM theorem. In: Proceedings of Symposia in Pure Mathematics (AMS). 69, 2001, pp. 707-732. ( Links to PDF file)
  • Rafael de la Llave: A tutorial on KAM theory. 2001, online copy.
  • Henrik Broer: KAM Theory - the legacy of Kolmogorov's 1954 paper. Bulletin of the American Mathematical Society, 2004
  • Celestial mechanics
  • Set ( mathematics)
  • Theory of dynamical systems
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