This paper gives a unified account of exact solutions for ellipsoids of uniform density. An approximate solution for a slightly spheroidal system of non-uniform density is given in the final section.

In computer experiments on the dynamics of stellar systems special methods are often required for the computation of the forces to keep the problem manageable. For collisionless systems a method based on solving Poisson's equation with Fourier-series expansions has been used with success (Miller et al., 1970; Hohl, 1973). The systems in these studies consist of a large number of particles moving on a rectangular grid.

Self-consistent dynamical models of elliptical galaxies have been constructed using a modification of the Prendergast-Tomer algorithm. They reproduce many of the observed properties of elliptical galaxies.

The properties of conservative dynamical systems of two or more degrees of freedom are reviewed. The transition from integrable to ergodic systems is described. Nonintegrability is due to the interaction of two, or more, resonances. Then one sees, on a surface of section, infinite types of islands of various orders, while the asymptotic curves from unstable invariant points intersect each other along homoclinic and heteroclinic points producing an apparent ‘dissolution’ of the invariant curves. A threshold energy is defined separating near integrable systems from near ergodic ones. The possibility of real ergodicity for large enough energies is discussed. In the case of many degrees of freedom we also distinguish between integrable, ergodic, and intermediate cases. Among the latter are systems of particles interacting with Lennard-Jones interparticle potential. A threshold energy was derived, which is proportional to the number of particles. Finally some recent results about the general three-body problem are described. One can extend the families of periodic orbits of the restricted problem to the general three-body problem. Many of these orbits are stable. An empirical study of orbits near the stable periodic orbits indicates the existence of 2 integrals of motion besides the energy.

Some conjectures are given about the sensitivity of the behaviour of the stellar orbits (appearance of resonant orbits, possible interaction of resonances etc.) to the set of parameters characterizing a simple realistic ‘spheroid + nucleus’ mass model of our Galaxy. Connections with a systematic exploration of orbits in Schmidt mass models are mentioned. Finally we present arguments in favour of the existence of a third integral for the whole solar neighbourhood old stellar populations, independently of the chosen model.

This paper reviews a series of investigations of the orbits of stars in the regions of the Lindblad resonances of a spiral galaxy. The analysis is formulated in an epicyclic approximation. Analytic solutions of the epicyclic equations of motion are obtained by the method of harmonic balance of Bogoliubov and Mitropolsky. These solutions represent the resonance phenomena exhibited by the orbits in generally excellent agreement with numerical solutions.

It is argued that in a galaxy like ours a third integral of motion, a third independent argument in the distribution function, should exist if the potential function has to satisfy a third condition imposed on it, namely symmetry with respect to a plane. Orbit computations of single stars in a symmetric potential of the kind (Martinet and Hayli, 1971) indicate that a third integral seems to exist for Population I stars while it ceases to exist for Population II objects. This situation is explained by the author as follows. We state that a third integral should exist for all populations alike if the Boltzmann equation is interpreted within the statistical context for which it is valid. When applied to a complex system like the galaxy the third integral of the Boltzmann equation, which holds for an elementary volume in phase space, will also hold for a single particle if the latter is representative of the behavior of the element of volume as in Population I (coherent motion) whereas it will not necessarily hold for a single star of Population II; in the latter population the elementary volume, containing the same number of stars, does not represent the behavior of the element of volume during the motion of this in phase space.

The ultimate aim of studies of the dynamics of stellar systems is to gain an understanding of why they have the structures that are observed, how they might have formed, and how they might evolve with time. In the case of galaxies, unlike smaller astronomical systems such as stars and star clusters, the problem of understanding their present structure is inseparable from the problem of understanding their formation since, as has long been clear, the two-body relaxation time in galaxies is much longer than the age of the universe, so that properties such as the radial density distribution and the flattening of elliptical galaxies cannot have changed significantly since they were formed. Other features of galaxies, such as the presence of spiral patterns or nuclear activity, may be of a more transitory nature and not closely related to the formation process, but such phenomena probably involve mainly the gas component of galaxies and may be regarded as perturbations on the underlying stellar system, whose basic structure must still be understood in terms of the formation process.

Simple collapse and violent relaxation models produce sharply cut off envelopes with ϱ(r) ∝ r–4, whereas actual elliptical galaxies have more extended envelopes approximately given by ϱ(r) ∝ r–3 as implied by Hubble's Law. Numerical models are presented, showing that when cosmological infall effects are included, galaxies are produced with more extended envelopes having a ϱ(r) ∝ r–2.8 dependence in excellent agreement with Hubble's Law. We have also computed a realistic rotating model of an E5 galaxy which includes infall effects and where the angular momentum of the galaxy is gained through tidal interactions with neighboring protogalaxies. The envelope displays a Hubble dependence but at great distances shows a cutoff due to the tidal effects. The model is compared to the E5 galaxy NGC 4697 and provides an outstandingly good fit to the observations. Except for simple scaling, this excellent fit is produced without the use of any free fitting parameters.

The study of gravitating systems of colliding particles has many potential astrophysical applications, for instance the dynamics of Saturn's ring, the formation of the solar system, the flattening of protogalaxies and the evolution of galactic nuclei. We consider numerically a three-dimensional system of particles moving in the gravitational field of a central mass point and interacting through inelastic collisions. After a very fast flattening, the system forms a disc of finite thickness: this disc spreads slowly, and collisions still occur. A central condensation is formed and there is an outward flux of angular momentum. The energy which is continually lost in the inelastic collisions is obtained at the expense of the bodies which fall inwards.

Current understanding of the stability of gas and stellar disks suggests very strongly that local stability to axisymmetric modes is not sufficient for global stability. A global instability to a bar mode will develop unless the rotational kinetic energy is sufficiently small compared with the random kinetic energy for the system as a whole. A disk as cool as the galactic disk near the Sun can survive only if most of the mass of the Galaxy is in a ‘hot’ component, such as a central bulge and/or an extended halo. We review the theoretical evidence for this conclusion coming from analytic results for simple gas and stellar disks, from numerical simulations of stellar disks, and from numerical calculations of the stability of gas disks. Some new results on the precise form of dynamic bar instabilities of gas disks with and without halos are reported.

Using a version of the Toomre-criterion (Toomre, 1964) for a mixture of stellar populations, limits are derived for the kinematical properties of the newly discovered population of low-velocity M-dwarfs (Weistrop, 1972; Murray and Sanduleak, 1972); these stars are claimed to be a large mass-fraction (≳ 0.5) of all stars in the solar neighbourhood and to have only 10 km s−1 velocity dispersion. They could make up for the locally missing mass (Oort, 1965; Schmidt, 1974). It is found that a configuration of stellar populations with the M-dwarfs included (having such properties) is not stable in the sense of the Toomre-criterion. The limiting stability properties are given.

The existence of a rich population of late M-type dwarfs with small peculiar velocities is discussed on the basis of the work of Weistrop, Murray and Sanduleak, Pesch and others. These stars probably supply the ‘missing’ mass in the solar neighborhood. Their small peculiar velocities create problems related to the stability of the galactic disk.

Currently available data on rotation curves are reviewed. For curves derived from optical measurements the distribution of the ratios: the last measured point on a rotation curve to the optical radius of the galaxy has a median value of if Reference Catalogue radii are used and if Holmberg radii are used. It is the absence of easily measurable H II regions that so severely limits the extent of these rotation curves. Accordingly, little can be said of the dependence of Vc on R for large R, where R is comparable to a Holmberg radius. The assumption that a rotation curve approaches a Keplerian curve after passing its peak rotational velocity implies a strongly concentrated and limited extent of the mass distribution within a galaxy. This assumption is not supported by 21-cm observations of the velocity field within a galaxy. Because of the greater extent of H I compared to measurable optical (blue) surface brightness, rotation curves may be defined to much larger radii from 21-cm observations. The median value of the above ratio for 14 galaxies is 1.3. At least 7 of these galaxies show an essentially constant rotational velocity at large R, while 5 galaxies have a slowly decreasing Vc(R). For both types of curves, a significant surface mass density at large R is required, and a large (≳ 100) mass-to-luminosity ratio is indicated. Such values are consistent with a late dwarf M star population (the most common type of star in the solar neighborhood) in the outer regions of a galaxy.

Models of galactic disks based on optical photometry and uniform M/L ratios have rotation curves which are indistinguishable from those derived from 21 cm hydrogen line observations.

The methods used in the large-scale n-body simulations are discussed. However, the present review concentrates on the results already obtained in the n-body simulations using systems containing up to 200000 simulation stars. Results are presented which show that the stability criterion developed for flattened systems applies only to truly axisymmetric instabilities. Purely stellar disks acquire rather large velocity dispersions, generally two or more times the velocity dispersion required by Toomre for axisymmetric stability. In the computer simulations, the bar-forming instability can be prevented only by comparatively large velocity dispersions. However, simulations inclusing the effects of the galactic halo and core as a fixed background field show that bar formation can be prevented for fixed halo components as large or larger than the self-consistent disk component. Experiments performed to determine the collisional relaxation time for the large-scale gravitational n-body calculations show that these models are indeed ‘collisionless’.

We discuss some observational aspects of the structure and dynamics of disk galaxies. Topics include the disk, the bulge, the lens, and the z-structure of the disk.

An attempt has been made to compute the angular momenta of the early type galaxies NGC 4697 and NGC 4762. Comparison with spiral galaxies indicates that there is a correlation between angular momentum per unit mass and morphological class.

This is a review of the observational evidence concerning compact supermassive objects, their formation and evolution, and their dynamical interaction with dense stellar systems in galactic nuclei.

All these have never yet been seen -
But Scientists, who ought to know,
Assure us that they must be so…
Oh! let us never, never doubt
What nobody is sure about!
Hilaire Belloc, 1910.

It is shown that massive objects can be ejected symmetrically in two opposite directions from the nucleus of a giant elliptical galaxy only under special initial conditions. A way of obtaining these conditions is outlined, and the resulting picture of evolution of galactic nuclei, including a quasar or a Seyfert galaxy phase, is shown to be compatible with observations.