### 3.4 Newtonian equations of motion for extended bodies

Before ending this section, we present some equations for Newtonian extended bodies (stars). These
equations will give a useful guideline when we develop our formalism.
The basic equations are the equation of continuity, the Euler equation, and the Poisson equation,
respectively:

We define the mass, the dipole moment, the quadrupole moment, and the momentum of the star as
Here is a representative point of the star . The time derivative of the mass vanishes. Setting the
time derivative of the dipole moment to zero gives the velocity momentum relation and a definition of the
center of mass,
where . Using the velocity momentum relation, we calculate the time derivative of the
momentum,
where is defined by Equation (43). The Newtonian potential can be expressed by the mass and
multipole moment as
Substituting into Equation (56), we obtain the equations of motion,
Here we ignored the mass multipole moments of the stars that are of higher order than the quadrupole
moments.
Actually, it is straightforward to formally include all the Newtonian mass multipole moments in the
surface integral approach,

where , , is a corrective index, denotes the
symmetric-tracefree operation on the indices between the brackets, and are the Newtonian mass
multipole moments of order .