GEOMECH
Geometric Mechanics
| Coordinatore | UNIVERSITEIT GENT
Organization address
address: SINT PIETERSNIEUWSTRAAT 25 contact info |
| Nazionalità Coordinatore | Belgium [BE] |
| Totale costo | 158˙400 € |
| EC contributo | 158˙400 € |
| Programma | FP7-PEOPLE
Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) |
| Code Call | FP7-PEOPLE-2009-IRSES |
| Funding Scheme | MC-IRSES |
| Anno di inizio | 2011 |
| Periodo (anno-mese-giorno) | 2011-01-01 - 2014-12-31 |
Partecipanti
| # | ||||
|---|---|---|---|---|
| 1 |
UNIVERSITEIT GENT
Organization address
address: SINT PIETERSNIEUWSTRAAT 25 contact info |
BE (GENT) | coordinator | 34˙200.00 |
| 2 |
AGENCIA ESTATAL CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS
Organization address
address: CALLE SERRANO 117 contact info |
ES (MADRID) | participant | 102˙600.00 |
| 3 |
OSTRAVSKA UNIVERZITA V OSTRAVE
Organization address
address: DVORAKOVA 7 contact info |
CZ (OSTRAVA 1) | participant | 21˙600.00 |
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Obiettivo del progetto (Objective)
'Geometric mechanics is the common name that is given to those research activities that are devoted to the application of differential geometry to various fields of theoretical physics, in particular to classical mechanics (Lagrange and Hamilton mechanics), dynamical systems theory and control theory. This proposal concerns a joint exchange programme in this broad field, centered around the following four topics: (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. There are three European partners involved: Ghent University (Belgium), Consejo Superior de Investigaciones Cientificas (Madrid, Spain) and the University of Ostrava (Ostrava, Czech Republic), and four non-European partners: Waseda University (Tokio, Japan), the University of California (San Diego, USA), Universidad Nacional del Sur (Bahia Blanca, Argentina) and La Trobe University (Bundoora, Victoria, Australia). The partners have been chosen both in function of a strong overlap of interests and in complementarity of expertise in the subfields covered by the project. The main objective consists in establishing new and reinforcing existing collaborations in the field of geometric mechanics through the exchange of both experienced and non-experienced researchers.'
Introduzione (Teaser)
Geometric mechanics exploits the fact that both geometry and symmetry principles underly most physical laws. An EU-funded research network pursued modern applications sharing the same concepts of symmetry and geometry.
Descrizione progetto (Article)
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' (http://www.geomechnetwork.ugent.be/ (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 http://arxiv.org/ (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.