Arnold diffusion

For integrable systems, one has the conservation of the action variables. According to the KAM theorem if we perturb an integrable system slightly, then many, though certainly not all, of the solutions of the perturbed system stay close, for all time, to the unperturbed system. In particular, since the action variables were originally conserved, the theorem tells us that there is only a small change in action for many solutions of the perturbed system.

However, as first noted in Arnold's paper,[1] there are nearly integrable systems for which there exist solutions that exhibit arbitrarily large growth in the action variables. More precisely, Arnold considered the example of nearly integrable Hamiltonian system with Hamiltonian

${\displaystyle H(I,p,q,\phi ,t)={1 \over 2}I^{2}+p^{2}+\epsilon \cos {(q-1)}+\mu \cos {(q-1)}(\sin {\phi +\cos t)}}$

He showed that for this system, with any choice of ${\displaystyle \epsilon ,\delta ,K>0,}$ where ${\displaystyle K\gg \delta }$, there exists a ${\displaystyle \mu _{0}>0}$ such that for all ${\displaystyle \mu \in (0,\mu _{0})}$ there is a solution to the system for which

${\displaystyle I(0)<\delta {\text{ and }}I(T)>K}$

for some time ${\displaystyle T\gg 0.}$

A background on the KAM theorem can be found in [3] and a compendium of rigorous mathmetical results, with insight from physics, can be found in.[4]

• KAM theorem
• Nekhoroshev estimates