Geometrisation of physical theories
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 mechanical systems. In the study of systems of particles and rigid bodies, the presence of symmetries and their associated invariants plays a crucial role. Since the middle of the last century, classical mechanics has evolved hand in hand with booming areas of mathematics such as differential geometry, the theory of symplectic and Poisson manifolds and Lie groups. The aim of the EU-funded project GEOMECH (Geometric mechanics) was to further advance the geometric modelling of physical theories. Geometric structures underlying these theories have been studied. Scientists from seven countries applied the tools and language of modern geometric mechanics and geometric control theory to investigate, for instance, mechanical systems that have rolling wheels. 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 their velocities. During the course of the GEOMECH project, the scientists shared their knowledge on such non-holonomic systems to deepen current understanding of their often counter-intuitive behaviour. They worked together on various discretisation schemes adapted to account for constraints. Geometrical integrators were constructed for systems with constraints that are affine in the velocities. GEOMECH scientists also treated the effects of symmetries in classical field theories. Continuous symmetries are mathematically represented by Lie group actions. These were used to reduce the number of degrees of freedom of the system on which they act by grouping together equivalent states and exploiting conserved quantities. Furthermore, a variational principle, called the Hamilton-Pontryagin principle, was introduced in the framework of classical field theories. GEOMECH scientists showed that the resulting implicit field equations can be described by an extension of the concept of Dirac structures. Specifically, they can be described using a graded version of Dirac structures. Progress was also made in the study of time-dependent mechanical systems that can be described as a special case of field theories. Research on the differential geometric analysis of second-order differential equations included the inverse problem of the calculus of variations. The latter deals with the problem of whether a system of differential equations is equivalent to a Lagrangian system. The many results of the GEOMECH project have been described in more than 90 papers published in or submitted to peer-reviewed journals. Collaborative work between the partner institutions brought forth new ideas supporting mathematical sciences research in Europe. It is hoped that established links will be further strengthened beyond the end of the project.
Keywords
GEOMECH, geometric mechanics, geometric control theory, classical field theories, calculus of variations
