The bi-Hamiltonian geometry of integrable systems

This is the text of a talk given in Dalmine on May 9, 2007, during one of the “scientific meetings” of the Faculty of Engineering of the Bergamo University. Since the aim of these meetings is to bring together scientists working in different fields, this presentation was intended to be understandable also for non-mathematicians. Using the example of the open Toda lattice, we give a very brief account of the history of integrable systems and we present the main ideas of a geometric approach based on the notion of bi-Hamiltonian system.