By Andre Avez

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TULCZYJEW 26 References. [1] R. E. Marsden, Foundations of Mechanics, Benjamin-Cummings (1978). [2] S. M. Tulczyjew, The geometrical meaning and globalization of the Hamilton-Jacobi method, Lecture Notes in Mathematics, 836 (1980), pp. 9-21. [3] J. M. Tulczyjew, A symplectic framework for field theories, Lecture Notes in Physics, 107, Springer-Verlag (1979). [4] A. Lichnerowicz, C. R. Acad. Sci. Paris, 280 (1975), pp. 37-40. R. M. Tulczyjew, Infinitesimal symplectic re lations and generalized Hamiltonian dynamics, Ann.

On the domain U of a natural chart, we set C = +B C 2q|U U U 1 where has exactly the type (t',1) (with t f ^ 2q+l) and C y the maximum order (t -l). We have : 2q+l where Ĺ q q has an order has the maximum order (2q+l). If C n 2q+l > (2q+l) it follows from calculus of S. Gutt concerning the Che s tie m n a aitr valley 2-cocycles that the main part of C^q+i ^ ^ P of a Chevalley 2-cocycle and so of 8A, where A is a differential 45 Deformations and Quantization operator. ^ has an order ^: 2q+l ^: t 3c 2q+l 2q+l 2q+l 2q+l T * ß ^q+l in u,w.

T. , . ) îkj If we choose T defined by : I (5-1) T. = (V, F. + V. , ) ijk 3 k ij J ik we see immediately that VF = 0 for Ă and that Ă is thus a sym plectic connection. A symplectic manifold admits infinitely many symplectic connections ; such two connections differ by a tensor of type (1,2) deduced from an arbitrary symmetric covariant 3~tensor. Moreover, if G is a Lie group acting on (W,F) by symplectomorphisms and if there is on (W,F) a linear connection invariant F) α symplectic connection invariant under under G, there is on (W3 G .