The response of a galaxy initially in equilibrium to a gravitational interaction with a companion is described by the simultaneous solution of the Boltzmann and Poisson equations. The simultaneous system is a set of coupled partial integro-differential equations. However if the orbits in each component are regular, any phase-space quantity--such as density and gravitational potential--may be expanded in a Fourier series in the orbital frequencies. Truncating this expansion, the quantity may be represented as a vector of Fourier coefficients; this is standard practice in filtering and approximation theory and canonical perturbation theory (e.g. Lichtenberg & Lieberman 1983). In Fourier space, the Boltzmann PDE becomes an algebraic integral equation. The system is further simplified if the basis functions are chosen to satisfy the Poisson equation explicitly. After a Laplace transform in time, the remaining solution of the Boltzmann equation becomes the solution of a matrix equation, each column describing the response to a particular basis function. See references cited abouve for mathematical detail.
To give the flavor of its use, consider two interacting galaxies.
Denote the expansion of the perturbation potential caused by the
companion galaxy as vector 
. Then the direct response of the
galaxy, vector 
may be written:
The matrix 
, the response operator, implicitly contains
the time-dependence of the perturbation as well as the dynamics of the
Boltzmann equation. If we are interested in the self-gravitating
response to the perturbation, we need to solve:
In words, equation (2) states that the self-consistent
reaction of a galaxy to a perturbation is the gravitational response
to both the perturbing force and the force of the response itself.
Equations (1) and (2) result from the Laplace
transform of the Boltzmann equation and therefore represent a
particular frequency component, 
. Therefore, the
solution of equations (1) or (2) requires an
inverse Laplace transform to recover its explicit time dependence (see
Paper II for details). See Nelson & Tremaine (1997)
for a general discussion of the response operator.