KAMtheorie

KAM-Theorie, or Kolmogorov–Arnold–Moser theory, is a cornerstone of Hamiltonian dynamics. It concerns nearly integrable systems, whose unperturbed version is integrable and whose phase space is filled by invariant Lagrangian tori carrying quasi-periodic motion. The theory shows that many of these tori survive small perturbations, leading to long-term stability for a large class of initial conditions.

Origin and statement: Kolmogorov introduced the persistence of most invariant tori for sufficiently smooth or analytic Hamiltonians with a non-degenerate frequency mapping. Later, Vladimir Arnold and Jörg Moser refined and completed the proofs in the 1960s. The KAM theorem requires smallness of the perturbation, a non-degeneracy (twist) condition, and a Diophantine condition on the frequencies of motion. Under these assumptions, a large measure set of invariant tori continue to exist and carry quasi-periodic dynamics, though their shapes may be deformed.

Limitations and scope: Tori that are resonant or fail the non-degeneracy conditions may disappear; the theory does not guarantee global stability. It is primarily applicable to finite-dimensional, analytic (or very smooth) Hamiltonian systems and to perturbations that are sufficiently small. There are extensions to infinite-dimensional systems and Hamiltonian partial differential equations, but these require additional hypotheses.

Impact and applications: KAM theory underpins the understanding of long-term stability in celestial mechanics and other nearly integrable systems. It also spurred developments in the study of dynamical systems, perturbation theory, and the analysis of PDEs via infinite-dimensional KAM results.