# Mechanics and differential geometry

Mathematical modelling of dynamical systems plays an important role in many branches of science, such as: physics, engineering, biology, chemistry, biotechnology, biochemistry and economics. While the complexity of realistic models often requires the use of numerical techniques and the help of a computer for determining their solutions, the abstract mathematical model serves to unravel and comprehend the qualitative features of the problem and its interaction with the real world.

Since the second half of last century, differential geometry has developed into a mathematical discipline with an ever growing impact on the construction of such models. In particular, "geometric mechanics" has become the common name that is given to those research activities that are devoted to the application of differential geometry in various fields of theoretical physics such as classical mechanics (Newtonian, Lagrangian and Hamiltonian mechanics), continuum mechanics, dynamical systems theory, control theory and quantum mechanics.

The underlying ideas of geometric mechanics go back to the principles of optics as formulated by Galileo, Descartes, Fermat and Huygens, and have gradually evolved and infiltrated in other parts of physics, in particular mechanics. Building on the works of Lagrange, Poisson, Hamilton, Lie and Poincare, among others, geometric mechanics really broke through in the middle of last century with the rise of modern differential geometry. Using the powerful tools and techniques of Riemann geometry, contact geometry, symplectic and Poisson geometry, and exploiting the properties of Lie groups, fibre bundles, jet bundles, connections, distributions, etc., geometric mechanics has contributed a lot to the description and analysis of the structure and properties of mechanical systems. In addition, many of these geometrical ideas have found an extension to field theories, classical and quantum, such as general relativity, classical and quantum gauge theories. Even for the developments in quantum gravity, such as string theory, differential geometry turns out to be an indispensable mathematical framework.

# The irses network geomech

The aim of this joint exchange programme is to establish or reinforce a collaboration between partners with a common interest in certain aspects of the above 'geometrization of physical theories'. The network is financially sponsored within Marie Curie's International Research Staff Exchange Scheme (irses) in the 7th European Framework Program, under project nr 246981. It will run from January 1, 2011 until December 31, 2014. # Work packages

The project will be centered around the following four general items

1. Geometric structures in mechanics and field theory;

2. Nonholonomic mechanics and geometric control theory;

3. Geometry of second-order differential equations and the calculus of variations;

4. Geometric integration techniques.