  Building on the development of a Hermite-Legendre analysis of one-dimensional
gravitating collisionless systems, we present a technique for determining the
steady states of such systems. This provides an important component for
understanding the physics involved in the relaxation of these kinds of systems.
As the dark matter structures in the universe should have traits in common with
these systems, insight into this relaxation can provide clues to larger
astrophysical questions. For large perturbation strengths, we determine
physically motivated parameter ranges for the simplest families of steady
states as well as their stability. We also demonstrate that any set of initial
conditions in the linear regime can be resolved into unique time-independent
and time-dependent modes. Combinations of time-independent modes then describe
the steady state of any system linearly perturbed from equilibrium. These
results highlight the importance of initial conditions over relaxation
mechanisms in the evolution of these systems.
