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Concept
Kolmogorov–Arnold–Moser theorem
Summary
The Kolmogorov–Arnold–Moser (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 Jürgen Moser in 1962 (for smooth twist maps) and Vladimir Arnold in 1963 (for analytic Hamiltonian systems), and the general result is known as the KAM theorem.
Arnold originally thought that this theorem could apply to the motions of the Solar System or other instances of the n-body problem, but it turned out to work only for the three-body problem because of a degeneracy in his formulation of the problem for larger numbers of bodies. Later, Gabriella Pinzari showed how to eliminate this degeneracy by developing a rotation-invariant version of the theorem.
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.
The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, i.e. there is a map from the original manifold to the deformed one that is continuous in the perturbation. Conversely, other invariant tori are destroyed: even arbitrarily small perturbations cause the manifold to no longer be invariant and there exists no such map to nearby manifolds. Surviving tori meet the non-resonance condition, i.e., they have “sufficiently irrational” frequencies.
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