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

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

The first three terms of this Hamiltonian describe a rotator-pendulum system. Arnold showed that for this system, for any choice of ${\displaystyle I_{+}>I_{-}>0}$, and for ${\displaystyle 0<\mu \ll \epsilon \ll 1}$, there is a solution to the system for which

${\displaystyle I(0)<I_{-}{\text{ and }}I(T)>I_{+}}$

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

His proof relies on the existence of `transition chains' of `whiskered' tori, that is, sequences of tori with transitive dynamics such that the unstable manifold (whisker) of one of these tori intersects transversally the stable manifold (whisker) of the next one. Arnold[1] conjectured that "the mechanism of 'transition chains' which guarantees that nonstability in our example is also applicable to the general case (for example, to the problem of three bodies)."

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

In Arnold's model the perturbation term is of a special type. The general case of Arnold's diffusion problem concerns Hamiltonian systems of one of the forms

(1) ${\displaystyle H_{\epsilon }(I,\phi ,p,q)=H_{0}(I,p,q)+\epsilon H_{1}(I,\phi ,p,q,t)}$

where ${\displaystyle (I,\phi ,p,q,t)\in \mathbb {R} ^{m}\times \mathbb {T} ^{m}\times \mathbb {R} ^{n}\times \mathbb {T} ^{n}\times \mathbb {T} ^{1}}$, ${\displaystyle m,n\geq 1}$, and ${\displaystyle H_{0}(I,p,q)}$ describes a rotator-pendulum system, or

(2) ${\displaystyle H_{\epsilon }(I,\phi )=H_{0}(I)+\epsilon H_{1}(I,\phi ,t)}$

where ${\displaystyle (I,\phi ,t)\in \mathbb {R} ^{N}\times \mathbb {T} ^{N}\times \mathbb {T} ^{1}}$, ${\displaystyle N\geq 2.}$

For systems as in (1), the unperturbed Hamiltonian possesses smooth families of invariant tori that have hyperbolic stable and unstable manifolds; such systems are referred to as a priori unstable. For system as in (2), the phase space of the unperturbed Hamiltonian is foliated by Lagrangian invariant tori; such systems are referred to as a priori stable.[5] In either case, the Arnold diffusion problem asserts that, for `generic' systems, there exists ${\displaystyle \rho >0}$ such that for every ${\displaystyle \epsilon >0}$ sufficiently small there exist solution curves for which

${\displaystyle \|I(T)-I(0)\|\geq \rho }$

for some time ${\displaystyle T\gg 0.}$ Precise formulations of possible genericity conditions in the context of a priori unstable and a priori stable system can be found in,[6][7] respectively. Informally, the Arnold diffusion problem says that small perturbations can accumulate to large effects.

Recent results in the a priori unstable case include,[8][9][10][11][12] and in the a priori stable case.[13][14]

In the context of the restricted three-body problem, Arnold diffusion can be interpreted in the sense that, for all sufficiently small, non-zero values of the eccentricity of the elliptic orbits of the massive bodies, there are solutions along which the energy of the negligible mass changes by a quantity that is independent of eccentricity.[15][16][17]

• KAM theorem
• Nekhoroshev estimates