Description
The Kalnajs matrix method is an important example of the application
of basis functions in galactic dynamics. It has been used to
determine the global response of a stellar system to perturbations in
frequency space, particularly in understanding the modes and
instabilities of stellar discs. There are, however, two shortcomings
of the method. First, it relies on the use of action-angle variables,
which greatly restricts the systems to which it can easily be
applied. Second, it is not trivial to transform from frequency space
into the temporal domain, which is natural if one wishes to
compare a linear response calculation to N-body simulations.
We show how these problems can be addressed by recasting the
linearised collisionless Boltzmann equation as an explicitly
time-dependent Volterra integral equation and applying Kalnjas' basis
function idea to that. We explain how the response kernel can
easily be constructed by direct orbit integration, avoiding
the need for angle-action variables and greatly increasing the range of problems to
which linear response theory can be applied. An important unsolved
problem is the choice of basis functions that makes these calculations
tractable in practice.
