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2003

5 December 2003
Rubén Martín Grillo, Department of Applied Mathematics IV, UPC, Barcelona
Vector hulls of jet bundles. An application to implicit differential equations
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 11 h: Abstract: Given a fibre bundle M --> B, it is known that the jet bundles J^kM --> J^k-1M are affine bundles. On the other hand, every affine bundle A can be canonically inmersed in a vector bundle Â over the same base space, called the vector hullof A (see the talk below).
In this talk, we will show how the vector hull of some jet bundles can be identified with well-known vector bundles. For example, when the base space is R, the tangent bundle TM is a useful model for the vector hull of J¹M. These results can be applied to the study of differential equations. In particular, given a time-dependent differential equation of a certain class, we show a method to obtain an equivalent autonomous one.

14 November 2003

Xavier Gràcia, Department of Applied Mathematics IV, UPC, Barcelona

Some fine points in affine geometry

Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 11 h

Abstract: For every affine space A there exists a canonical immersion A --> Â as a hyperplane in a vector space, which could be called the vector hull of A. This little known fact is greatly clarifying, both for affine geometry and for its applications.
Expressions like a+u=b and u=b-a, or barycentric calculus, have a neat interpretation inside Â, and the projective completion of A is nothing but P(Â). The linear representation of GA(n,K) as a subgroup of GL(n+1,K) is also well-known. On the other hand, working in Â allows to devise elegant algorithms to draw curves and surfaces for computer-aided geometric design.
In the field of differential geometry, jet manifolds provide with examples of affine bundles. In particular, the jet space J¹M of a bundle M --> R is an affine bundle over M. One can define an affine immersion of J¹M in the tangent bundle TM, and it turns out that this bundle is a model of the vector hull of J¹M.

The purpose of this talk is to study some affine-geometric and algebraic aspects of the vector hull, and to show that an apparently quiet domain as affine geometry can hide pleasant surprises.

A brief motivation
A universal problem
A construction of the vector hull
Review of the bibliography, other constructions and their equivalence

References

Y. Bamberger et J.-P. Bourguignon, "Torseurs sur un espace affine", pp. 151--202 in L. Schwartz, Les tenseurs, Hermann, Paris, 1975.
M. Berger, Géométrie 1, Nathan, 1990 (first edition published in 1977).
N. Bourbaki, Algèbre, châpitres 1-3, Hermann, Paris, 1970.
J. Gallier, Geometric methods and applications for computer science and engineering, Springer-Verlag, New York, 2001.
E. Martínez, T. Mestdag and W. Sarlet, "Lie algebroid structures and lagrangian systems on affine bundles", J. Geom. Phys. 44 (2002) 70--95.
L. Ramshaw, "Blossoms are polar forms", Digital Equipment Corporation, research report, 1989.

20 May 2003
Demeter Krupka, Institute of Theoretical Physics, Masaryk University, Brno
Variational principles for energy-momentum tensors
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 11:30 h: Abstract: It is well known that field equations, derived from a variational principle, possess two different notions of energy-momentum tensors: the first one based on symmetries of the underlying variational functional (the Noether-like energy-momentum tensor), and the second one obtained from a variational extension of the field equations (variational energy-momentum tensors). In the present talk, relations of variational energy-momentum tensors to the inverse problem of the calculus of variations on fibered manifolds are discussed.

19 May 2003
Olga Krupková, Mathematical Institute, Silesian University at Opava
Recent results in Hamiltonian field theory, I: Hamiltonian systems associated with Euler-Lagrange equations, Lepagean and multisymplectic forms
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 16.00 h

20 May 2003
Olga Krupková, Mathematical Institute, Silesian University at Opava
Recent results in Hamiltonian field theory, II: Regular variational problems and regularization of Lagrangians
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 16.00 h: Abstracts: Classical Hamilton-De Donder equations are related with the Poincaré-Cartan form of a Lagrangian. For Lagrangians satisfying a regularity condition they are equivalent with the Euler-Lagrange equations, and it is possible to find a coordinate transformation (Legendre transformation) bringing the Hamilton-De Donder equations to a canonical form. It turns out, however, that almost all interesting Lagrangians in field theory (e.g. Dirac field, electromagnetic field, gravity, Yang-Mills fields) do not satisfy the regularity condition, hence there is not a clear Hamiltonian counterpart to their Euler-Lagrange equations.
In the present two talks we shall discuss recent generalizations of Hamilton theory for first and higher order variational problems on fibered manifolds, based on the concept of Lepagean (n+1)-form. These forms are closed counterparts of Euler-Lagrange forms, and provide Hamilton equations associated directly with Euler-Lagrange equations (i.e. the same for the class of equivalent Lagrangians). Since, in this setting, Hamilton equations become equations for integral sections of a differential system, it is possible to understand the concepts of regularity and of Legendre transformation geometrically as properties of the Hamiltonian differential system. As a result, one obtains not only a geometric meaning for these concepts, but also generalized regularity conditions and Legendre transformation formulas.
It is also significant that regularity conditions depend on a concrete choice of a Lepage form associated with the given variational problem (in this context, the Poincaré-Cartan form represents one possible choice) -- this leads to a regularization procedure for Lagrangians. We shall show that all the above mentioned physical field Lagrangians are regularizable, and have appropriate Hamilton equations, Hamiltonian, and independent momenta.
Lepage (n+1)-forms and the arising Hamilton equations are closely related with multisymplectic forms: these relations will be also discussed.

7 May 2003
Rafael Ramírez, Department of Computer Engineering and Mathematics, Universitat Rovira i Virgili, Tarragona
Cartesian approach for nonholonomic systems
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 16 h: Abstract: In the history of mechanics, there have been two point of views for studying the problem of dynamics: the Newtonian (the movements of celestial bodies can be described by differential equations of the second order) and the Cartesian (the movements can be described by equations of the first order).
In this talk we develop the Cartesian approach to describe nonholonomic systems. We apply the obtained results to study the classical system with 3 degrees of freedom.

18 March 2003
Rubén Martín Grillo, Department of Applied Mathematics IV, UPC, Barcelona
Time-dependent singular systems
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 10.15 h: Abstract: This talk deals with time-dependent linearly singular systems. This geometric object is suitable for the study of singular differential equations which are affine in the velocities. We describe a constraint algorithm in order to find the solutions of these systems. The formalism is applied to model singular systems which arise from mechanics.

17 January 2003
Jorge Cortés Monforte, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign
Characterization of gradient control systems
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 10.15 h: Abstract: We investigate necessary and sufficient conditions under which a generalnonlinear affine control system with outputs can be written as a gradient control system corresponding to some pseudo-Riemannian metric defined on the state space. The results rely on a suitable notion of compatibility of the system with respect to a given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role in the discussion.