Editing Kolmogorov–Arnold–Moser theorem (section)

==Statement==

===Integrable Hamiltonian systems===

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 condition]]s of the integrable Hamiltonian system will trace different invariant [[torus|tori]] in phase space.  Plotting the coordinates of an integrable system would show that they are quasiperiodic.

===Perturbations===

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. 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). The KAM theorem quantifies the level of perturbation that can be applied for this to be true.

Those KAM tori that are destroyed by perturbation become invariant [[Cantor set]]s, named ''Cantori'' by [[Ian C. Percival]] in 1979.<ref>{{Cite journal|title = A variational principle for invariant tori of fixed frequency|journal = Journal of Physics A: Mathematical and General|date = 1979-03-01|pages = L57–L60|volume = 12|issue = 3|doi = 10.1088/0305-4470/12/3/001|first = I C|last = Percival|bibcode = 1979JPhA...12L..57P }}</ref>

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.

As the perturbation increases and the smooth curves disintegrate we move from KAM theory to Aubry–Mather theory which requires less stringent hypotheses and works with the Cantor-like sets.

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.

===Consequences===

An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.{{which|date=January 2016}}