We first exchange the order of integrals,
The reason we introduced is as follows. Suppose that we treat an integrand for which the super-potential is available. By calculating the Poisson integral, we have a piece of field corresponding to the integrand. The piece generally depends on , however we reasonably discard such -dependent terms (other than logarithmic dependence) as explained in Section 4.5. Using the so-obtained -independent field, we evaluate the surface integrals in the general form of the 3 PN equations of motion by discarding the dependence emerging from the surface integrals, and obtain the equations of motion. Thus the “discarding-” procedure must be employed each time when the field is derived and also when equations of motion are derived, not just once. Thus was introduced to distinguish the two kinds of dependence and to discard the dependence in the right order. We show here a simple example. Let us consider the following integral:
With this caution in mind, it is straightforward (though tedious) to evaluate the surface integrals and then the volume integrals, and thus evaluate all the necessary integrals for our derivation of the 3 PN equations of motion.
When we have derived the 3 PN equations of motion, we have used the super-potential method whenever possible, and used a combination of the above three methods when necessary. In fact, for a computational check, we have used the direct-integration method to evaluate the contributions to the equations of motion from all of the 3 PN nonretarded field, and . As expected, we obtain the same result from two computations; the result from the direct-integration method agrees with that from the combination of the three methods: the direct-integration method, the super-potential method, and the super-potential-in-series method.
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