A quantitative Birkhoff Normal Form for the hinged-hinged beam equation with geometric nonlinearity

TL;DR

This paper develops a precise analytical method to understand small amplitude solutions of a nonlinear hinged-hinged beam system over large time scales, using a quantitative Birkhoff Normal Form approach.

Contribution

It introduces a novel quantitative Birkhoff Normal Form for the nonlinear beam equation, enabling detailed long-term analysis of small solutions.

Findings

Effective integration of the system with small residuals

Optimized estimates for realistic physical parameters

Enhanced understanding of long-term behavior of nonlinear beams

Abstract

We consider an undamped nonlinear hinged-hinged beam with stretching nonlinearity as an infinite dimensional hamiltonian system. We obtain analytically a quantitative Birkhoff Normal Form, via a nonlinear coordinate transformation that allows us to integrate the system up to a small reminder, providing a very precise description of small amplitude solutions over large time scales. The optimization of the involved estimates yields results obtained for realistic values of the physical quantities and of the perturbation parameter.