“First Integrals and Normal Forms for Nonlinear Oscillators”
Giovedì 4 Aprile 2024, ore 14:30 - Aula 2AB40 - Antonio Giorgilli (Università degli Studi di Milano)
Abstract
A revisitation of the old standing problem of constructing first integrals for a nonlinear system of oscillators. Equivalently: the dynamics in a neighborhood of an elliptic equilibrium point. After Poincare’, who proved that in general an integrable Hamiltonian system subject to a small perturbation does not possess a first integral independent of the Hamiltonian, attention has been focused on elliptic equilibria. I will describe two methods: a direct one (not often exploited) which consists in solving by series the equation for a first integral; and an indirect one (more common) which exploits the transformation of the Hamiltonian into a normal form, for example formally integrable. The formal solution, however, raises the question of investigating the (non) convergence of the formal series so constructed. This entails a discussion of the impact of small divisors. I will try to shed some light on this problem, thus identifying the mechanism that leads to divergence of the formal series.
