Chapter 13 started out by noting the importance of the study of orbits as a key step in the study of the dynamics of galaxies. Here the reason why it is important is briefly summarized. Even better than other stellar systems, elliptical galaxies may be modeled as purely collisionless collections of stars. It is well known that for the distribution function in phase space that is used to describe these systems, the evolution operator dictated by the mean field is the same as that for singleparticle orbits. The resulting Jeans theorem (see Chapter 9) thus reduces the problem of finding the most general stationary solution to the collisionless Boltzmann equation to identification of the relevant integrals of the motion. Furthermore, the frequencies characterizing single-particle orbits play an important role in the stability of the stellar system with respect to internal or driven perturbations. Finally, we should recall that closed periodic orbits are often used to model relatively smooth and cold flow patterns that are sometimes observed around galaxies, thus offering a very interesting diagnostic tool for the underlying potential (see also Subsection 13.6.1 and the discussion of dark matter in Chapters 20 and 24). Unfortunately, when we deal with the dynamics of elliptical galaxies and with the related orbits, we have to face two very difficult issues.

The first important point is that ellipticals are dynamically hot stellar systems. Therefore, even if we focus our attention on axisymmetric basic states, approximations such as those that have led to the discussion of quasi-circular orbits in the context of spiral galaxies are bound to be of little use.