# Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

## TL;DR

This paper proves the existence of Arnold diffusion in nearly integrable Hamiltonian systems with arbitrary degrees of freedom, using geometric and variational methods to show orbits that traverse significant action space along resonances.

## Contribution

It extends Arnold diffusion results to the a priori stable case with arbitrary degrees of freedom, employing new geometric and variational techniques.

## Key findings

- Existence of diffusion orbits in a priori stable systems.
- Diffusion occurs along co-dimension one resonances.
- Finite set of additional resonances are the only obstructions.

## Abstract

We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) + $\epsilon$H1($\theta$, p, t), $\theta$ $\in$ T n , p $\in$ B n , t $\in$ T = R/T be a nearly integrable system of arbitrary degrees of freedom n 2 with a strictly convex H0. We show that for a "generic" $\epsilon$H1, there exists an orbit ($\theta$, p)(t) satisfying p(t) -- p(0) {\textgreater} l(H1) {\textgreater} 0, where l(H1) is independent of $\epsilon$. The diffusion orbit travels along a co-dimension one resonance , and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination geometric and variational methods, and manage to adapt tools which have recently been developed in the a priori unstable case.