January 11-13, 2010 - Ghent (Belgium)
Activity
Introduction:
This meeting in Ghent is the fourth in a series of workshops that previously have taken place in Madrid (2006, 2007) and Barcelona (2008). Its goal is to bring together young researchers working in geometric mechanics and control theory and to offer a platform to present the results of their research to an international audience. Attendance is, of course, open to anyone, but in particular young researchers (PhD-students, recent PhD's) are encouraged to deliver a talk at this meeting. The following is a non-exhausted list of topics that can be covered during the workshop:
- Geometry: (multi)symplectic geometry, Poisson and Jacobi manifolds, Lie groups, Lie algebroids and Lie groupoids.
- Mechanics: Lagrangian and Hamiltonian systems, non-holonomic mechanics, calculus of variations, mechanical systems with symmetry, conservation laws and reduction, classical field theories, geometric integration of mechanical systems, geometric quantization.
- Control: optimal control theory, control of mechanical systems, geometric control.
Courses:
Expository talks by
- Silvere Bonnabel (Ecole des Mines de Paris): Observers and symmetries, theory and examples Observers are real-time model-based estimators which allow to filter the measurement noise, and to estimate some quantities which are not directly measured. For nonlinear systems, there is no general method to design observers. In this talk, we propose to focus on a special class of nonlinear systems : systems possessing symmetries (or invariances). For those systems we propose a general method to build candidate observers which respect the symmetries. For instance the conservation laws involved in the dynamic models of chemical reactors are independent of the choice of physical units (mol, kg, ...). Thus it seems logical that the estimation algorithms do not depend on the units either. In this case, the method allows to find all observers which are invariant to a change of units. Symmetry-preserving observers also have very interesting properties. Indeed, in the particular case of a left-invariant dynamics on a Lie group, the (estimation) error equation is autonomous. This is a strong property reminding the linear autonomous case. We will apply this method to the example of multisensor fusion for mobile vehicles, in which the symmetries correspond to Galilean invariances. The theory of symmetry-preserving observers only deals with the cases where the dimension of the symmetry group is smaller than the dimension of the state (for instance the group acts by changing a physical unit). In the end of the talk we will mention the case where the transformation group consists of all changes of coordinates (infinite-dimensional transformation group). Indeed, we will consider conservative Lagrangian systems with configuration (position) measurements. We will present a globally convergent intrinsic velocity-observer. To do that, we will use the Jacobi metric, and we will view the Lagrangian dynamics as a geodesic flow on a manifold. This work is based on the paper by Aghannan and Rouchon (IEEE-TAC, 2003).
- Oscar Fernandez (University of Michigan): The Hamiltonization of Nonholonomic Systems and its Applications It has been known since the early 1900's that although nonholonomic systems are not Hamiltonian, they can nonetheless be put into a Hamiltonian form via a suitable time reparameterization (a process called Hamiltonization). In this talk I will discuss this procedure, along with recent work extending the basic theorem in the field (the Chaplygin Reducibility Theorem). In addition, I will discuss the many applications that arise if a nonholonomic system is Hamiltonizable. In particular, the interesting idea of applying variational integrators to these non-variational systems will be discussed, as will the equally interesting but somewhat more challenging question of the quantization of a nonholonomic system. Time permitting, I will also discuss new work relating Hamiltonizable systems to the alternate (though entirely equivalent) formulation of (vacuum) general relativity known as teleparallel gravity.
- Francois Gay-Balmaz (Caltech): Semidirect products and cocycles in classical mechanics In this expository talk, I will first review the principal properties of semidirect products with cocycles. This is a large class of group extensions that includes both semidirect products and central extensions. When used in the context of Hamiltonian and Lagrangian reductions, these group extensions form a powerful tool that enables us to study complex systems and to formulate new models. I will present in detail some recent results obtained in the context of superfluids, nematic liquid crystals, Vlasov statistical moments, and molecular strands experiencing nonlocal interactions.
- Bavo Langerock (St-Lucas Institute, Ghent): Routh reduction for quasi-invariant Lagrangians We first recall the classical formulation of Routh reduction as a reduction procedure for Lagrangian systems with cyclic coordinates. We generalize this to Lagrangian systems with a quasi-cyclic coordinate: these are systems for which the Lagrangian is invariant under translations in a coordinate but only up to a total time derivative of a certain function. We show how a reduction technique described in [1] called `functional Routhian reduction' can be seen as a particular instance of this quasi-cyclic Routh reduction. The second part of the talk concerns the generalizaton of this reduction procedure towards actions of arbitrary non-abelian Lie groups. We proceed in a more general differential geometric framework and describe Routh reduction as a special case of standard symplectic reduction. [1] Ames, A. D., Gregg, R. D. and Spong, M. W.``A geometric approach to three-dimensional hipped bipedal robotic walking'' In 46th IEEE Conference on Decision and Control, pp. 5123-5130 (2007).
- Ari Stern (University of California, San Diego): Variational integrators and symplectic geometry This expository lecture will introduce the theory, techniques, and applications of variational integrators for geometric mechanics. These numerical integrators are developed by discretizing the action principle of Lagrangian mechanics, and consequently they automatically preserve several important geometric structures. We will also discuss the connection between variational integrators and the canonical symplectic structure of the cotangent bundle, particularly with respect to generating functions and Lagrangian submanifolds. Finally, time permitting, we will also survey some of the most recent and ongoing research developments in this field.
Schedule:
Monday, January 11 | Tuesday, January 12 | |
9.00-10.15 | Bonnabel | Gay Balmaz |
10.15-10.45 | Serres | Urban |
10.45-11.15 | Coffee | Coffee |
11.15-12.30 | Fernandez | Stern |
12.30-13.00 | Balseiro | Jiménez |
13.00-14.30 | lunch | lunch |
14.30-15.00 | Dahamna | Garcia-Naranjo |
15.00-15.30 | Ghezzi | Cronstrom |
15.00-16.00 | Coffee | Coffee |
16.00-16.30 | Vilariño | Bucataru |
16.30-17.00 | Yoshimura | de Nicola |
17.00-17.30 | Campos | Adlaj |
19.30 | Conference dinner |
Wednesday, January 13 | |
9.00-10.00 | Langerock |
10.00-10.30 | Tronci |
10.30-11.00 | Coffee |
11.00-11.30 | Ricardo |
11.30-12.00 | Shunjie |
Participants:
- Semjon Adlaj (CCRAS, Moscow)
- Paula Balseiro (IMPA, Rio de Janeiro)
- Maria Barbero (Queen's University)
- Silvere Bonnabel (Ecole des Mines de Paris)
- Ioan Bucataru (University "Al.I.Cuza", Iasi)
- Cedric Campos (CSIC, Madrid)
- Frans Cantrijn (Ghent University)
- Christofer Cronstrom (University of Helsinki)
- Khaled Dahamna (Institut national des sciences appliquées de Rouen)
- Javier de Lucas (Polish Academy of Sciences)
- Antonio de Nicola (University of Coimbra)
- Oscar Fernandez (University of Michigan)
- Luis Garcia Naranjo (École Polytechnique Fédérale de Lausanne)
- Francois Gay-Balmaz (Caltech)
- Roberta Ghezzi (International School for Advanced Studies, Trieste)
- Xavier Gracia (Technical University of Catalonia)
- David Iglesias-Ponte (CSIC, Madrid)
- Fernando Jimenez (CSIC, Madrid)
- Bavo Langerock (St- Lucas Institute, Ghent)
- Juan Carlos Marrero (Universidad de La Laguna)
- David Martin de Diego (CSIC, Madrid)
- Tom Mestdag (Ghent University)
- Edith Padron (Universidad de La Laguna)
- Miguel Rodriguez-Olmos (University of Manchester)
- Sandra Ricardo (University of Trás-os-Montes e Alto Douro)
- Willy Sarlet (Ghent University)
- David Saunders (University of Olomouc)
- Nicola Sansonetto (University of Verona)
- Ulysse Serres (Université Lyon 1)
- Li Shunjie (Institut national des sciences appliquées de Rouen)
- Diana Sosa (Universidad de La Laguna)
- Ari Stern (University of California at San Diego)
- Martina Stolarova (University of Ostrava)
- Cesare Tronci (École Polytechnique Fédérale de Lausanne)
- Zbynek Urban (Palacky University in Olomouc)
- Joris Vankerschaver (Ghent University)
- Silvia Vilarino (University of A Coruña)
- Goedele Waeyaert (Ghent University)
- Hiroaki Yoshimura (Waseda University)
- Juraj Zatko (University of Ostrava)