Invariant distributions and collisionless equilibria

This paper discusses the possibility of constructing time‐independent solutions to the collisionless Boltzmann equation which depend on quantities other than global isolating integrals such as energy and angular momentum. The key point is that, at least in principle, a self‐consistent equilibrium can be constructed from any set of time‐independent phase‐space building blocks which, when combined, generate the mass distribution associated with an assumed time‐independent potential. This approach provides a way to justify Schwarzschild’s method for the numerical construction of self‐consistent equilibria with arbitrary time‐independent potentials, generalizing thereby an approach developed by Vandervoort for integrable potentials. As a simple illustration, Schwarzschild’s method is reformulated to allow for a straightforward computation of equilibria which depend only on one or two global integrals and no other quantities, as is reasonable, for example, for modelling axisymmetric configurations characterized by a non‐integrable potential.