Our aim is to develop a general approach for the dynamics of material bodies of dimension d represented by a mater manifold of dimension (d + 1) embedded into the space-time. It can be declined for d = 0 (pointwise object), d = 1 (arch if it is a solid, flow in a pipe or jet if it is a fluid), d = 2 (plate or shell if it is a solid, sheet of fluid), d = 3 (bulky bodies). We call torsor a skew-symmetric bilinear map on the vector space of affine real functions on the affine tangent space to the space-time. We use the affine connections as originally developed by Élie Cartan, that is the connections associated to the affine group. We introduce a general principle of covariant divergence free torsor from which we deduce 10 balance equations. We show the relevance of this general principle by applying it for d from 1 to 4 in the context of the Galilean relativity.