The principle of general covariance, as formulated by V. Guillemin and S. Sternberg, is used to derive the passive equations of motion for a spinning matter field in the presence of an external gravitational field with torsion. In the case where the matter field becomes concentrated along a curve, the equations of motion for a spinning particle studied by D. Rappaport and Sternberg are recovered. The results of A. Einstein, L. Infeld, and B. Hoffmann and of J. M. Souriau are thus generalized to include spin and torsion. [@? 1985 Academic Press. Inc. A general scheme using symplectic geometry was presented in [12] deriving the equations of motion for classical particles in the presence of given external fields. In [8] these equations were studied for the special case of a spinning particle in the presence of curvature and torsion. In the present paper we obtain the equations of [S] by an entirely different method -the principle of general covariance as formulated by Souriau in [9] and generalized in [3, 10, 41. The present article can be read independently of the preceding papers.
We begin by recalling the principle of general covariance. One starts with a (generally infinite dimensional) group, 3, which represents the group of symmetries of all possible physical systems under consideration. In the case of classical general relativity (without torsion) one takes +? to be the group of all diffeomorphisms of the four manifold, M, with compact support. (A diffeomorphism has compact support if it agrees with the identity outside a compact set.) In our case we shall choose 9 as follows: Let H be the orthogonal group for some vector space V with a nondegenerate scalar product. (For the physical application we would take V = R', 3 to be Minkowski space so H would be the Lorentz group. But our considerations work in general.) Let P, -+K M be an abstract principal bundle, with structure group H. In particular, right multiplication of elements of P, by elements of H makes sense. A diffeomorphism 9 of P, is called an automorphism of P, if &Pa