The themes introduced in Chapter 10 are demonstrated in one class of dynamical systems, which, because of its simplicity, serves as a bridge from the case of single-particle dynamics to the more complex collective behavior of continuous media. The study of incompressible self-gravitating fluids with ellipsoidal shape focuses on an environment with a limited number of degrees of freedom. Still it opens the way to several subtle phenomena that have attracted the interest of many eminent mathematicians and physicists, starting with Newton.

The classical ellipsoids were originally studied as models, as a starting point, to interpret the properties of rotating celestial bodies, such as planets and stars. Curiously, similar concepts have recently found applications in completely different fields, such as the physics of atomic nuclei and black holes. Tools related to the simple properties of ellipsoidal mass distributions have found applications in the study of the formation of structures by collapse (“pancakes” or protogalaxies) in the cosmological context.

In galactic dynamics, classical ellipsoids might be thought of as the natural representation of elliptical galaxies. Unfortunately, spectroscopic observations in the mid-1970s proved that bright ellipticals are not flattened by rotation and thus do not fit this simple picture (see Subsection 4.2.2 and Part IV).