Let the function computing the derivative of
be .
*f* has no dependence on time. The Jacobian
of *f*, is

where is the 3*N*-square zero matrix, and *I* is the 3*N* identity
matrix. If all particles move, then the entire Jacobian is 6*N* 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 6*M* square. The only difficult part of this Jacobian is
, which is a 3*N*-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 .

Sun Dec 29 23:43:59 EST 1996

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