In this course we study integrable mechanical systems from a geometric point of view, using concepts and techniques that yield insight on small perturbations away from integrability and allow for generalization towards infinite dimensions.Lagrangian and Hamiltonian systems naturally occur in frictionless mechanics, but are also used to answer questions that arise in optics or when studying certain partial differential equations. After a short introduction we present the necessary theory motivated by examples like the geodesic flow or normal form approximations of non-integrable systems. The course ends with a bi-Hamiltonian (algebraic) approach to infinite-dimensional systems.
A central result is the existence of so-called action angle variables of an integrable Hamiltonian system. The action variables are conserved quantities in involution, which becomes particularly important in the infinite-dimensional case. Expressing a given system in action angle variables not only simplifies the study of the dynamical properties of the system itself, but also makes perturbations of the system accessible to both quantitative and qualitative investigations. See also http://www.math.uu.nl/people/hansmann/gm.html