KAMteori

Kolmogorov–Arnold–Moser (KAM) theory is a central result in dynamical systems that describes the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations. Developed in the 1950s and 1960s by Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser, it provides a rigorous mechanism for understanding long-term stability in systems such as celestial mechanics.

In its typical form, one studies a Hamiltonian H(I, θ) = H0(I) + ε H1(I, θ), where I are action variables and θ are angle variables, with ε a small parameter. The unperturbed part H0(I) yields invariant tori on which motion is quasi-periodic with frequency ω(I) = ∂H0/∂I. KAM theory shows that, for sufficiently small ε, most invariant tori with non-resonant (Diophantine) frequencies survive the perturbation, though they may deform. The surviving tori form a Cantor-like set of positive measure, and the motion on each torus remains quasi-periodic with adjusted frequencies. The result is established via an iterative, near-identity canonical transformation that eliminates non-resonant perturbations, under conditions of non-degeneracy of H0, smoothness (analyticity or high differentiability), and a Diophantine condition on ω.

Limitations include breakdown near resonances, the emergence of chaotic regions, and the need for small ε and sufficient smoothness. KAM theory has profoundly influenced celestial mechanics, the study of stability in Hamiltonian dynamics, and extensions to partial differential equations and dissipative systems, shaping our understanding of stability and long-term evolution in complex dynamical systems.