A nonlinear Coulomb interaction term appears in the equations describing the
motions of a collection of Paul-trapped particles. A realistic description may also include a damping term which arises from collisions with a background gas
and/or from laser cooling. The motion of the 
ion in a collection of
N ions is given by:
where 
, 
, and 
are the
damping forces in the x-,
y-, and z- directions respectively, 
is the distance between
particles j and i, and 
is the unit vector pointing from
particle j to particle i.