On the algebraic properties of exponentially stable integrable hamiltonian systems

TL;DR

This paper develops algebraic conditions involving derivatives up to fifth order to determine steepness, a key geometric property ensuring long-term stability in near-integrable Hamiltonian systems, advancing the theoretical understanding of their stability.

Contribution

It introduces algebraic criteria for steepness based on derivatives up to fifth order, extending previous geometric approaches and paving the way for more general stability conditions.

Findings

Algebraic conditions for steepness involving derivatives up to fifth order.

Insights into the genericity of steepness in Hamiltonian systems.

First step towards conditions involving higher-order derivatives.

Abstract

Steepness is a geometric property which, together with complex-analyticity, is needed in order to insure stability of a near-integrable hamiltonian system over exponentially long times. Following a strategy developed by Nekhoro-shev, we construct sufficient algebraic conditions for steepness for a given function that involve algebraic equations on its derivatives up to order five. The underlying analysis suggests some interesting considerations on the gener-icity of steepness and represents a first step towards the construction of sufficient conditions for steepness involving the derivatives of the studied function up to an arbitrary order.