By D. Arnal (auth.), M. Cahen, M. De Wilde, L. Lemaire, L. Vanhecke (eds.)

This quantity includes the textual content of the lectures that have been given on the Differential Geometry assembly held at Liege in 1980 and on the Differential Geometry assembly held at Leuven in 1981. the 1st of those conferences used to be extra oriented towards mathematical physics; the second one has a more robust flavour of research. The Editors are happy to thank the lectures who contributed scientifically to those conferences. also they are thankful to Professor M. F1ato who has inspired booklet of those contributions within the Mathematical Physics stories sequence. We additionally thank the F.N.R.S. who supported financially the touch crew in differential geometry. The Universite de Liege and the Katholieke Universiteit Leuven that have given us a hot hospitality have contributed to the good fortune of those conferences. We convey our gratitude. The Editors. M. Caken et al. (6ds.), Differential Geametry and Mathematical Physics, vii. vii Copyright e 1983 by way of D. Reidel Publishing corporation. Lectures given on the assembly of the Belgian touch team on Differential Geometry held at Liege, could 2-3,1980 SIMULTANEOUS DEFORMATIONS OF A LIE ALGEBRA AND ITS MODULES D. Arnal college of Dijon advent We divulge the following a few effects that are bought via a workforce on the college of Dijon. This group incorporated Jean-Claude Cortet, Georges Pinczon and myself.

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**Additional info for Differential Geometry and Mathematical Physics: Lectures given at the Meetings of the Belgian Contact Group on Differential Geometry held at Liège, May 2–3, 1980 and at Leuven, February 6–8, 1981**

**Sample text**

1) by the complete galilean group(8) , or else in the relativistic case, by the Poincare group(9). The action of these groups on U by symplectomorphisms is defined in a natural way if the dynamical system is isolated; otherwise one considers a partial system, to which the "mechanism" made up by the given exterior system leaves only the symmetry corresponding to a subgroup of the galilean (or Poincare) group. 1); but also a centrifugal machine, etc. 1). It is sufficient to achieve this to consider all one-parame- ter subgroups of G; each one will be characterized by an element Z of the Lie algebra G of G; to each one will correspond a hamiltonian which will be denoted by M(Z).

All fits solutions satisfy both principles of thermodynamics and admit a detailed balance (energy-impulse, momentum). c. it contains in particular all equilibrium situations of statistical mechanics, and also the relativistic theory of elasti- city. d. Finally its non-relativistic limit allows one to identify the usual thermodynamic variables and in particular it contains the theory of elasticity, the mechanics of perfect fluids, the theory of heat conduction (Fourier) and the theory of viscosity (Navier).

Now we choose a compact K of space-time E4 (see fig. II) wherein we perturb the guv . The new space of motions u' is still a symplectic manifold, which can be connected to U by the technique of diffusion; this technique will be described in the case of a spinless particle, whose motion is characterized by the world line; if this line does not meet K, it characterizes a motion equally in U as in u'. Consider now a motion in U, which we shall denote by Xin whose world line centers at some time into K; with the perturbed poten- tials, the line will deviate from the initial motion (dotted line in figure).