By F. J. Bloore (auth.), P. L. García, A. Pérez-Rendón, J. M. Souriau (eds.)

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**Example text**

For r > 2 the proof is analogous. 6) is an equality for r > I. 2 does not imply a globalization of the construction of spaces Ar for r > I. Not having found a coplete intrinsic characterization of these spaces we resort to a technique of globalization based on coordinate invariance. 3. 1). 10) for each p E ¢ " P Proof. Let 2' and 8' be derivations associated with the chart (t'~). a m The action of 2' on ~-,0 is determined by its action on elements f E ~0,0 and dvf E t1,0. ,0. 16) am and ~a n am for n ~ e a.

Let D be the union considered in the proposition. If ix,y)e D, we can take a sequence (a,5 ..... ~) such that (x,y)gD~ o D# .... o D . , D'~TC6, whose union is an integral manifold of D'. Hence (x,y) 6 D and D ~ D . Conversely, if (x,y)~ D, we take an integral curve of D' with end points x and y and cover it by a finite sequence C~ ,Ca ..... i ~ ~ for i= 1 ..... r-1 (compare with the proof of Proposition 3). ,r-l. This means that = (x 4 , x r ) e D4 o Da . . ¢-- D . ,r-l. Then D ~ D. D. In some cases it may be necessary to consider complete solutions in the generalized sense (Morse families); for example when C does not project onto an open submanifold of N.

ON THE PREQUANTUM DESCRIPTION OF SPINNING PARTICLES IN AN EXTERNAL GAUGE FIELD C. DUVAL Facult~ des Sciences de Luminy et Centre de Physique Th~orique CNRS Marseil]e (~) Introduction. In the last few years there has been a renewed interest in the so called problem of the equations of motion. ascribed to the recent achievements physicists and geometers test particles The reason for this must be in gauge theories which prompted to tackle the question of the dynamics of in an external Yang-Mills field.