Destruction of Lagrangian torus for positive definite Hamiltonian systems

TL;DR

This paper demonstrates that in certain Hamiltonian systems, Lagrangian tori can be destroyed by small perturbations, contrasting with the persistence of KAM tori under slightly larger perturbations, highlighting differences in stability.

Contribution

The paper proves the destruction of Lagrangian tori with a fixed rotation vector under small perturbations in positive definite Hamiltonian systems, contrasting with KAM torus persistence results.

Findings

Lagrangian tori can be destroyed by arbitrarily small $C^{2d- ext{delta}}$-perturbations.

KAM tori with constant frequency persist under $C^{2d+ ext{delta}}$-small perturbations.

The stability thresholds differ for Lagrangian and KAM tori in these systems.

Abstract

For an integrable Hamiltonian $H_0=1/2\sum_{i=1}^dy_i^2$ $(d\geq 2)$, we show that any Lagrangian torus with a given unique rotation vector can be destructed by arbitrarily $C^{2d-\delta}$-small perturbations. In contrast with it, it has been shown that KAM torus with constant type frequency persists under $C^{2d+\delta}$-small perturbations.