moments of an arbitrary, isolated system of bodies in the form of integrals over the compact supports of the bodies, and, they explicitly checked that the lowest moments were conserved quantities. Our task here will consist in transforming their expressions, involving integrals over the global-time simultaneity surface, into a combination of terms involving integrals over $N$ separate local-time simultaneity surfaces. Notice that nothing guarantees a priori that these manipulations will lead to final expressions containing only the “good” moments introduced in the DSX scheme. In fact, we shall succeed in this task for seven of the globally conserved quantities and fail for three of them (the three components of the trickier total angular momentum).

Let us note in advance that there are several limiting cases for which it is already known that some of the globally conserved quantities can be entirely expressed in terms of some dynamically relevant individual multipole moments. First, there are the cases where one truncates the multipolar series: keeping only the monopole contributions defines the “Lorentz-Droste-Einstein-Infeld-Hoffmann” model (LD-EIH), while keeping also the intrinsic spins of the bodies define the “pole-dipole” model (PD). In both cases, we dispose of well established forms of the conservation laws (see 