Let the function computing the derivative of 
be 
.
f has no dependence on time. The Jacobian
of f, 
is
where 
is the 3N-square zero matrix, and I is the 3N identity
matrix. If all particles move, then the entire Jacobian is 6N square.
If only M particles move, then there is no meaning to the entries in
the Jacobian owned by particles that do not move, and so the Jacobian is
only 6M square. The only difficult part of this Jacobian is
, which is a 3N-square matrix, and
where 
is the acceleration of particle i, given by equation
A.1,
and 
is its position in 3-space. Thus,
Finally, if we let 
represent one of 
, it
can be shown that
where, for example, 
is the 
component of 
.
If softening is employed, substitute 
for 
and 
for 
.
Wayne Hayes