MASIE: Mechanics and symmetry in Europe: the geometry and dynamics of deformable systems

• Research unit: EU RFP
• Project number: 00.0490

Key words (English)

Mechanics; geometry; Hamiltonian system; stability; bifurcation; periodic orbit; relative equilibrium;
Education; Training; Scientific Research; Social Aspects

Alternative project number (English)

EU project number: HPRN-2000-00113

Research programs (English)

EU-programme: 5. Frame Research Programme - 4.1.1 Research training networks

Short description (English)

See abstract

Further information (English)

Full name of research-institution/enterprise:
EPF Lausanne
Faculté des Sciences de base
Section de Mathématiques, Institut Bernoulli - Bât. MA

Partners and International Organizations (English)

Coordinator: University of Warwick (UK)

Abstract (English)

In the period 01/06/01 --31/05/02 work on Hamiltonian symmetric bifurcation theory was carried out. Juan-Pablo Ortega has left this group for a permanent position at the Institut Nonelineaire de Nice and became part of the MASIE team there. However, the MASIE projects in which he was involved remained active and collaborative work has been carried out on a regular basis.
The main thrust of the work in this period was on singular reduction and the so-called optimal momentum map. In a Hamiltonian system with symmetry, there are, under certain conditions, conserved quantities that generalize the classical linear and angular momentum. Noether's Theorem holds. However, there are other preserved objects that are invisible to the momentum map, such as the symmetry type of the points in phase space. In addition, there are canonical actions that do not admit a momentum map. A major generalization of the momentum map has been proposed that is able to perceive such quantities and it exists for any canonical action. Its properties have been studied. The reduced spaces relative to this optimal momentum map are the leaves of the singular reduced spaces, if a standard momentum map exists. In addition, a slice theorem for canonical actions that do not necessarily admit momentum maps has been proved.
The study of second grade fluids has been another part of the work in this past year. Well posedness for the second grade fluid equations in two and three dimensions and global existence and uniqueness in two dimensions has been proved. Liapunov stability conditions of stationary solutions on two dimensions have been determined.
The MASIE graduate student, Razvan Tudoran has settled in this past year and has begun work on a problem emnating from the Caltech Ph. D. Thesis of Antonio Hernandez regarding the blow up of the amended potential and its use in classical bifurcation problems. So far he was already able to significantly improve Hernandez' result by being able to trat toral, as opposed to circular, symmetries and to eliminate a major nondegeneracy assumption. He is in the process of writing these results up.

References in databases (English)

Swiss Database: Euro-DB of the
State Secretariat for Education and Research
Hallwylstrasse 4
CH-3003 Berne, Switzerland
Tel. [phone]
Swiss Project-Number: 00.0490