Graduate School in Nonlinear Science

Sponsored by The Danish Research Agency

MIDIT                              OFD                          CATS
Modelling, Nonlinear Dynamics      Optics and Fluid Dynamics    Chaos and Turbulence Studies
and Irreversible Thermodynamics    Risø National Laboratory     Niels Bohr Institute and 
Technical University of Denmark    Building 128                 Department of Chemistry
Building 321                       P.O. Box 49                  University of Copenhagen 
DK-2800 Lyngby                     DK-4000 Roskilde             DK-2100 Copenhagen Ø
Denmark                            Denmark                      Denmark

Lecture 1. FOUNDATIONS OF CLASSICAL MECHANICS:

We start with reviewing the basic principles of Lagrangean and Hamiltonian mechanics, focusing mainly on geometrical and topological aspects. We discuss Poincare recurrence theorem, Huygens variational principle and the interconnection between mechanics and geometrical optics, simplectic structure, and some other topics.

Lecture 2. INTEGRABILITY AND NON-INTEGRABILITY IN HAMILTONIAN MECHANICS:

We begin with Arnol'd-Liouville thoerem on integrable systems and some classical results on existence of first integrals of motion. We give some new criteria for nonexistence of analytic first integrals, based on topological structure of the system configuration space. We then proceed to the theory of perturbations, introduction of KAM theory and the structure of near-integrable Hamiltonian systems.

Lecture 3. CONSTRAINED MOTION:

We discuss various aspects of mechanical systems with constraints and realisation of constraints. We give an introduction to Vakonomic mechanics and Dirac hamiltonian formalism with constraints. We show the connection of these problems to the classical optimisation and control problems.