**Programme**

**Monday**

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

**Tuesday**

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

**Wednesday**

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

**Thursday**

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

**Friday**

9:00–10:30 Serge Preston

10:30–11:00 Coffee break

11:00–12:30 Demeter Krupka

*Departure*

**Abstracts**

**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

- - jets, differential groups, differential invariants
- - bundles of frames, frame liftings, jet prolongations of frame liftings,
- - associated liftings

2. Variational calculus on fibred manifolds (introduction)

- - Lepage forms
- - The inverse problem of the calculus of variations
- - Hamilton theory: foundations
- - variationality and energy-momentum tensors

3. Natural Lagrange structures,

- - Euler-Lagrange equations
- - regularity and Hamilton equations
- - conservation laws problems

4. General relativity: fundamental Lepage form

- - Hilbert-Young-Mills functional