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2 December 2004
José Antonio Vallejo, Department of Applied Mathematics IV, UPC, Barcelona
Einstein's equations as a lagrangian dynamical system
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 12.00 h
Abstract   It is well known that Einstein's equations can be obtained through a variational procedure; in fact, this is the technique used in the very first derivation of the equations, due to Hilbert [4].   However, the common variational setting is not analogous to that of dynamical systems:  there is no dynamical entities involved (only an integration of the scalar curvature over the whole spacetime [6]). 
A true dynamical-variational approach must deal with the evolution of a geometrical structure, in this case a three dimensional metric, and this amounts to do differential geometry on the space of such structures.   Thus, the basics of differential calculus on spaces of mappings between manifolds and the techniques of variational calculus adapted to them (following a recent approach by Olga Gil-Medrano and myself [2,3,5]) will be exposed and then applied to the formulation of Einstein's equations as a lagrangian dynamical system.   Also, it will be shown that the results that arise in this way are complementary to those obtained by Fischer and Marsden [1] from the hamiltonian point of view.


  1. A. E. Fischer, J. E. Marsden: "The Einstein equations of evolution: a geometric approach". J. Math. Phys. 13 (1972) 546-568.
  2. O. Gil-Medrano: "Relationship between volume and energy of unit vector fields". Diff. Geom. Appl. 15 (2001) 137-152.
  3. O. Gil-Medrano: "Gradients and hessians of geometric functionals". Pages 65-77 in Proceedings of the IX Fall Workshop on Geometry and Physics, Vilanova i la Geltrú, 2000, Publicaciones de la RSME 3, Madrid, 2001.
  4. D. Hilbert: "Die grundlagen der Physik". Nach. Ges. Wiss. Götingen 21 (1915) 461-472.
  5. J. A. Vallejo: "Puntos críticos de funcionales definidos sobre espacios de aplicaciones entre variedades. Aplicación a las ecuaciones de Einstein como sistema dinámico". Research report, Dep. de Geometria i Topologia, Universitat de València, 2001.
  6. S. Weinberg: Gravitation and cosmology. Wiley, New York, 1972.

1, 2, 3, 4, 7 and 8 June 2004   Ph.D. course
Juan Carlos Marrero, Dept. of Fundamental Mathematics, University of La Laguna, Canary Islands
The rigid body and the Lie-Poisson reduction theorem
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 16.00 h
  1. Particle mechanics in generalizaed coordinates and the rigid body
  2. Poisson structures
  3. Basic aspects on Lie groups
  4. Lie-Poisson reduction theorem
  1. R. Abraham, J.E. Marsden: Foundations of Mechanics. Benjamin, 1978 (2nd ed).
  2. J.E. Marsden, T.S. Ratiu: Introduction to mechanics and symmetry. Springer Verlag, 1994.
  3. I. Vaisman: Lectures on the geometry of Poisson manifolds. Birkhauser, Basel, 1994.

11, 12, 13, 14, 17 and 18 May 2004   Ph.D. course
Giuseppe Marmo, Dept. of Physical Sciences, Università di Napoli "Federico II"
Some advanced topics on the geometry of dynamical systems
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 16.00 h
  1. Implicit differential equations for particle dynamics and field theories.
  2. Symmetries and conservation laws for implicit equations.
  3. Formal integrability: constraints and the Dirac-Bergmann formalism.
  4. Partial differential operators and symbols: from waves to trajectories.
  5. A C-star algebraic approach to dynamical evolution.
  1. Tullio Levi-Civita, Caratteristiche dei sistemi differenziali e propagazione ondosa. Zanichelli, Bologna, 1931.
  2. W.M. Tulczyjew, Geometrical Formulation of Physical Theories, Bibliopolis, Napoli, 1989.
  3. D.V. Alekseevskij, A.M. Vinogradov, V.V. Lychagin, Geometry I, Enciclopaedia of Mathematical Sciences vol. 28, Springer, New York, 1999.
  4. G. Marmo, E.J. Saletan, A. Simoni, B. Vitale, Dynamical Systems, John Wiley, Chichester, 1985.
  5. A.P. Balachandran, G. Marmo, B.S. Skagerstam, A.Stern, Gauge Symmetries and Fiber Bundles, Lecture Notes in Physics vol. 188, Springer, 1983. Classical Topology and Quantum States, World Scientific, Singapore, 1991.
  6. G. Esposito, G. Marmo, E.C.G. Sudarshan, From Classical to Quantum Mechanics, Cambridge University Press.

3 May 2004
Josep Llosa, Department of Fundamental Physics, University of Barcelona
On the degrees of freedom of a (semi)riemannian metric
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 16.30 h
Abstract   It is known, since an old result by Riemann, that an n-dimensional metric has f = n(n-1)/2 degrees of freedom, i. e., it is locally equivalent to the giving of f functions.   This feature seems to be a non-covariant property, for it is related to some particular choice of either local charts or local bases. 
In the 2-dimensional case, however, this Riemann result is intrinsic and covariant, i.e., only tensor quantities are involved, and the sole degree of freedom is represented by a scalar, conformal deformation factor, which only depends on the metric. 
The question thus arises of, whether or not, for n > 2 there exist similar intrinsic and covariant local relations between an arbitrary metric, on the one hand, and the corresponding flat one together with a set of f covariant quantities on the other. 
After realizing that n(n-1)/2 is precisely the number of independent components of an n-dimensional 2-form, in the context of the General Theory of Relativity, B. Coll has conjectured that any n-dimensional metric can be locally obtained as a parametrized deformation of a constant curvature metric, the parameter being a 2-form. 
Here we show the validity of Coll's conjecture for n = 4.

15 March 2004
Yuri Fedorov, Department of Applied Mathematics I, UPC, Barcelona
Nonholonomic LR Systems and their reduction to a Hamiltonian form
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 16.30 h
Abstract   There exists an ample variety of dynamical systems which are not (at least a priori) Hamiltonian, but possess an invariant measure.   This is a rather strong property which puts such systems close to Hamiltonian ones. 
We consider a class of systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as Chaplygin systems on the principal bundle G --> Q = G/HH being a Lie subgroup. It appears that the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. 
For G=SO(n) and a special choice of the left-invariant metric on SO(n), we prove that, after a time substitution, the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere Sn-1.   This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension.

9 January 2004
Jorge Cortés Monforte, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign
From geometric optimization and nonsmooth analysis to distributed coordination algorithms
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 12.15 h
Abstract   Consider n sites evolving within a convex polygon according to one of the following interaction laws:  (i) each site moves away from the closest other site or polygon boundary, (ii) each site moves toward the furthest vertex of its own Voronoi polygon, or (iii) each site moves toward a geometric center (centroid, circumcenter, incenter, etc) of its own Voronoi polygon.   These interaction laws give rise to strikingly simple dynamical systems whose behavior remains largely unknown.   Which are their critical points?   What is their asymptotic behavior?   Are they optimizing any aggregate function?   In what way do these local interactions give rise to distributed systems?   Are they of any engineering use in robotic coordination problems?   In this talk, we'll try to answer some of these questions.

9 January 2004
Sonia Martínez Díaz, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign
Scalable coordination algorithms for mobile sensing networks
Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 11.30 h
Abstract   This talk describes recent progress on motion planning and coordination problems for multi-vehicle systems.   The work is motivated by emerging applications in active sensor networks and autonomous robotic systems.   We study multi-vehicle networks performing spatially-distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor.   We propose gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies.   The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.   The proposed approach unifies concepts and methods from systems theory and robotics, resource allocation, nonsmooth and geometric optimization, and distributed algorithms.