The general astrophysical *N*-body system consists of *N* particles
moving according to Newton's three laws of motion, with Newton's familiar
gravitational law being the only source of force.

Let be the unit vectors of the standard
Cartesian (*x*,*y*,*z*) system.
Let be
the position vector of particle *i*.
Let be
the vector pointing from particle *i* to particle *j*, *i.e., *
the position of particle *j* with respect to particle *i*.
Thus

The force that particle *j* exerts on particle *i* is

where *G* is the gravitational constant, are the masses of
the particles, is the vector pointing from particle *i* to
particle *j*, is the magnitude of this vector, and
is the unit vector pointing in the direction of .
Let the total force on particle *i* be . It is the sum of all the
forces from all the other particles.
Thus the total force on particle *i* is

For a 3-dimensional space, this is a set of 3*N* second-order
ordinary differential equations (ODEs), which we translate into a set of
6*N* first-order ODEs by letting the velocity
and then building
the phase-space vector of the entire system as
.
Thus the first-order ODE system is

where is the vector representing the accelerations of all the particles.

At any given time, the position and velocity of every particle is known. Thus is simply , and is the set of time-derivatives of the velocity of each particle, where

If force softening is used, the denominator instead becomes where is the softening parameter.

Sun Dec 29 23:43:59 EST 1996

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