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.
[1] This was rigorously proved and extended by Jürgen Moser in 1962[2] (for smooth twist maps) and Vladimir Arnold in 1963[3] (for analytic Hamiltonian systems), and the general result is known as the KAM theorem.
Later, Gabriella Pinzari showed how to eliminate this degeneracy by developing a rotation-invariant version of the theorem.
[4] 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.
Surviving tori meet the non-resonance condition, i.e., they have “sufficiently irrational” frequencies.
This implies that the motion on the deformed torus continues to be quasiperiodic, with the independent periods changed (as a consequence of the non-degeneracy condition).
Those KAM tori that are destroyed by perturbation become invariant Cantor sets, named Cantori by Ian C. Percival in 1979.
[5] The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom.
As the number of dimensions of the system increases, the volume occupied by the tori decreases.
The existence of a KAM theorem for perturbations of quantum many-body integrable systems is still an open question, although it is believed that arbitrarily small perturbations will destroy integrability in the infinite size limit.
An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.[which?]
The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasiperiodic motions, now known as KAM theory.
case is normally excluded in classical KAM theory because it does not involve small divisors.