Exploring geometric mechanics

Two basic formulations of classical mechanics are those of Hamilton and Lagrange. These formulations are both elegant and general in the sense that they provide a unified framework for treating seemingly different physical systems, ranging from classical particles and rigid bodies to field theories and quantum systems. Since the middle of the last century, classical mechanics and classical field theories have evolved hand in hand with booming areas of mathematics such as differential geometry and the theory of Lie groups.

The aim of the EU-funded 'Geometric mechanics' (GEOMECH) project was to bring together scientists working on the 'geometrisation' of physical theories. They applied the tools and language of modern geometric mechanics to investigate, for example, mechanical systems that have rolling wheels without slipping and/or certain kinds of sliding contact. These systems are examples of so-called non-holonomic systems. Unlike classical Lagrangian or Hamiltonian systems, these more general systems are subjected to constraints on the velocities, and quite often they exhibit a counter-intuitive behaviour. In the context of the GEOMECH project, mathematicians from seven countries shared their knowledge on these non-holonomic systems and deepened the current understanding of their behaviour. Also the discretization of mechanical systems of non-holonomic type and the construction of numerical integrators for them have been studied.

GEOMECH scientists also treated the effect of symmetry in mechanics and field theory. Symmetries are mathematically represented by Lie group actions and they can be used to reduce the number of degrees of freedom of the system on which they act by grouping together equivalent states and exploiting the occurrence of conserved quantities.

A variational principle, called the Hamilton-Pontryagin principle, was introduced in the framework of classical field theory. The GEOMECH scientists showed that the resulting field equations can be described by an extension of the concept of Dirac structure.

Progress has also been made in the study of time-dependent mechanical systems, which were described as a special case of field theory, and on the differential geometric analysis of second-order differential equations, including the inverse problem of the calculus of variations. The latter deals with the problem of investigating whether or not a system of differential equations is equivalent to a Lagrangian system.

The close collaboration between GEOMECH partners resulted in more than 80 papers published in peer-reviewed journals or uploaded on arXiv. Links established with the research done by physicists provided a unique opportunity to bring forward new ideas supporting mathematical sciences research. It is hoped that joining their efforts will impact the future of geometric mechanics in Europe.