The Steep Nekhoroshev's Theorem

TL;DR

This paper offers a fully constructive, quantitative proof of Nekhoroshev's theorem for steep Hamiltonian systems, establishing explicit stability exponents based on system steepness indices.

Contribution

It revises Nekhoroshev's geometry of resonances and provides a new proof with explicit stability exponents for steep Hamiltonian systems.

Findings

Exponential stability exponent can be explicitly computed as 1/(2n α_1...α_{n-2})

Provides a constructive and quantitative proof of Nekhoroshev's theorem

Improves understanding of stability in steep Hamiltonian systems

Abstract

Revising Nekhoroshev's geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev's theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be $1/ (2n \alpha_1\cdots\alpha_{n-2}$) ($\alpha_i$'s being Nekhoroshev's steepness indices and $n\ge 3$ the number of degrees of freedom).