Kolmogorow-Arnold-Moser theorem

The Kolmogorow-Arnold-Moser theorem (“ KAM theorem ” for short ) is a result of the theory of dynamic systems , which makes statements about the behavior of such a system under small disturbances. The theorem partially solves the problem of the small divisors, which appears in the perturbation calculation of dynamic systems, especially in celestial mechanics .

The KAM theorem arose from the question of whether a small disturbance of a conservative dynamic system leads to a quasi-periodic movement. Andrei Kolmogorov achieved the breakthrough in answering this question in 1954 in his plenary lecture at the International Congress of Mathematicians in Amsterdam in 1954 (The general theory of dynamical systems and classical mechanics). The result was strictly proven in 1962 by Jürgen Moser for so-called smooth twist maps and in 1963 by Wladimir Arnold for Hamiltonian systems .

Heuristic

The main result of the KAM theory guarantees the existence of quasi-periodic solutions for a certain class of differential equations . The differential equations for the so-called n-body problem are an important subclass of this . Quasi-periodic solutions can be close to one another, but there can be unstable orbits between them, so that in practice, for example, due to finite measurement accuracy, it cannot be decided whether one is on a stable or unstable orbit. For the planetary system it can be shown that the unstable orbits are much rarer than the stable ones.

The theorem

If an undisturbed system is not degenerate, then most of the non-resonant tori are only slightly deformed for sufficiently small autonomous Hamiltonian perturbations , so that invariant tori also exist in the phase space of the disturbed system, which are wrapped around by the phase trajectories densely and quasi-periodically, whereby the frequencies are rationally independent. These invariant tori form the majority in the sense that the degree of complement of their union is small when the perturbation is weak.

literature

• Jürgen Pöschel: A lecture on the classical KAM-theorem . In: Proceedings of Symposia in Pure Mathematics (AMS) . 69, 2001, pp. 707-732. ( Links to the PDF file )
• Rafael de la Llave: A tutorial on KAM theory . 2001, online copy .
• Hendrik Broer : KAM-Theory - the legacy of Kolmogorovs 1954 paper . Bulletin American Mathematical Society, 2004
• H. Scott Dumas: The KAM story. A friendly introduction to content, significance and history of Kolmogorov-Arnold-Moser theory, World Scientific 2014
• Alessandra Celletti , Luigi Chierchia : KAM stability and celestial mechanics, Memoirs AMS 2007

Individual evidence

1. ^ Proc. Int. Congress Math. Amsterdam 1954, North Holland 1957, Volume 1, pp. 315–333 (Russian), English translation in Ralph Abraham, Jerrold E. Marsden, Foundations of Mechanics, Benjamin-Cummings 1978 (2nd edition), Appendix
2. ↑ Kolmogorow also published: On the conservation of conditionally periodic motions for a small change in Hamilton's function (Russian), Docl. Akad. Nauka SSSR, Volume 98, 1954, pp. 525-530, English translation Lecture notes in physics 93, 1975, pp. 51-56
3. ^ Moser, On invariant curves of area preserving maps of an annulus, Nachrichten Gött. Akad. Wiss., 1962, pp. 1-20
4. ^ Arnold, Proof of a theorem by AN Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Usp. Math. Nauka, Volume 18, 1963, pp. 13-40 or Russian Mathematical Surveys, Volume 18, 1963, pp. 9-36
5. ↑ Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys, Volume 18, 1963, pp. 85–191, corrections in Russian Uspekhi Mat. Nauk., Volume 23, 1968, p. 216