Total destruction of Lagrangian tori

TL;DR

This paper demonstrates that for certain Hamiltonian systems, all invariant Lagrangian tori can be completely destroyed using arbitrarily small analytic perturbations in a specific topology.

Contribution

It proves the total destruction of Lagrangian tori in integrable Tonelli Hamiltonians via small analytic perturbations, extending understanding of Hamiltonian stability.

Findings

All Lagrangian tori can be destroyed by small perturbations.

Destruction occurs in the $C^{d- ext{delta}}$ topology.

Applicable to systems with $d \, (d \, \geq 2)$ degrees of freedom.

Abstract

For an integrable Tonelli Hamiltonian with $d\ (d\geq 2)$ degrees of freedom, we show that all of the Lagrangian tori can be destroyed by analytic perturbations which are arbitrarily small in the $C^{d-\delta}$ topology.