16th International Summer School on

Global Analysis and Applications

Brno, Czech Republic, August 8–12, 2011



9:00–9:30 Registration
9:30–11:00 Serge Preston
11:00–11:30 Coffee break
11:30–13:00 Demeter Krupka
Free time
18:00–20:00 Opening drink

9:00–10:30 Serge Preston
10:30–11:00 Coffee break
11:00–12:30 Demeter Krupka
12:30–14:00 Lunch break
14:00 Individual Discussions

9:00–10:30 Demeter Krupka
10:30–11:00 Coffee break
11:00–12:30 Serge Preston
12:30–14:00 Lunch break
15:00–18:00 Facultative excursion

9:00–10:30 Demeter Krupka
10:30–11:00 Coffee break
11:00–12:30 Serge Preston
12:30–14:00 Lunch break
14:00 Poster Session and Individual Discussions
18:00–23:00 Conference party

9:00–10:30 Serge Preston
10:30–11:00 Coffee break
11:00–12:30 Demeter Krupka


Introduction to Non-Commuting Variations: Geometry and Applications in Physics.

Serge Preston

Our lectures will be devoted to the geometrical method of “non-commuting variations” (NC-variations) in Variational Calculus. NC-variations appeared first in the works on the non-holonomic mechanics of V.Volterra (1898) and L.Boltzman (1902). Under different names, the NC-variations appeared in the “Lezione de Mechanica Racionale” by Levi-Civita, Amaldi (1926), in “Analytical Mechanics” by A. Lurie (1961) and in “Dynamics of Non-Holonomic Systems” by Ju. Neimark and N.Fufaev (1972). In a series of works at 1972-1983, B.Vujanovich and T. Atanackovic applied the NC-variations to study the heat conduction in solids and other irreversible processes. In 1995-99, H. Kleinert and his collaborators used the non-holonomic transformations, essentially introducing the NC-variations in Mechanics in the spaces with Cartan geometry.

Recently, it has became clear that the methods of these researchers were actually based on the introduction of a non-trivial connection in the jet bundles over the configurational space of the physical systems. Using such (dynamical) connection, one can redefine the variations of the derivatives of dynamical fields by replacing standard total derivatives with covariant ones. As a result, Euler-Lagrange equations acquire the (non-potential) source terms.

We introduce the NC-variations for Lagrangian problems in field theory and discuss the role of the curvature of corresponding connections, Noether balance equations, Energy-Momentum balance law. We consider the construction of dynamical connection obtained by applying gauge transformation to the trivial connection. We study the types of dissipative sources that can be incorporated into the Euler-Lagrange equations using NC-variations.

As the applications of NC-variations we study the Euler-Lagrange equations of the Field Theory with a dissipative potential and the Lagrangian form of Thermoelasticity which delivers the Entropy Balance as one of Euler-Lagrange equations. Finally, we establish a relation between the entropy production and the rate of internal (material) time.

Exercises, provided during the lectures, will help the students to get acquainted with the geometry of non-commuting variations in Variational Calculus and their applications in Physics.

Variational Structure of Field Theory and General Relativity

Demeter Krupka

1. Differential invariants

2. Variational calculus on fibred manifolds (introduction)

3. Natural Lagrange structures,

4. General relativity: fundamental Lepage form