KAM tori are very sticky: rigorous lower bounds on the time to move away from an invariant Lagrangian torus with linear flow

Abstract

Hamiltonian perturbation theory is used to derive rigorous Nekhoroshev type lower bounds on the time it takes to move away from an invariant Lagrangian torus when the flow on the torus is conjugate to a highly non-resonant linear flow. In particular these results give bounds on the time to move away from a KAM torus. If r is the distance from the torus then the lower bound on the time to double the distance from the torus is (C/r)e ( Kγ r ) α , where C and K are constants, α ≡ 1/( n + 2), n is the number of degrees of freedom and γ is a measure of the irrationality of the flow. We thus prove that nearby trajectories take an extremely long time to move away from such an invariant torus, and that the more irrational the linear flow, the ‘stickier’ the torus will be. These estimates do not require the steepness or quasi-convexity assumptions of Nekherosev's theorem.