The key approximation of stellar dynamics is that stellar systems are collisionless, so the actual motion is well approximated by motion in a smooth potential. Orbits in smooth potentials are mostly quasiperiodic, and when they are not, it is possible to construct a nearby Hamiltonian in which the same initial conditions yield quasiperiodic motion. Therefore quasiperiodic motion is an excellent starting point for stellar dynamics.
Quasiperiodic orbits display an elegant structure that is captured by angle-action coordinates: each orbit is a three-torus. The angle variables specify where on its torus a star is, and they evolve linearly in time. If the frequencies are incommensurable (as they nearly always should be) a star's probability density should be independent of angle variables, so the distribution function depends only on the actions. The actions provide a geometrical quantification of the orbit.
Normally the true Hamiltonian will differ from the approximate one that admits angle-action variables. The small difference h between the true Hamiltonian and the approximate one can be important if it generates forces that act in the same way over extended periods of time. In the vicinity of resonances this may happen, and then h may change the dynamics qualitatively. The overall impact on the dynamics of a galaxy is nevertheless likely to be small if the resonance is isolated. When several resonances are simultaneously active, however, chaos can be generated, and stars may slowly diffuse through phase space. This process is likely to be important for the secular evolution of barred galaxies, but our understanding of it is currently inadequate.
The potential of a real galaxy is always fluctuating, and fluctuations are of fundamental importance because they alone permit stars to exchange energy. No matter what their physical origin, fluctuations will drive the system towards unattainable thermodynamic equilibrium, especially by enhancing core-halo structure. Two-body relaxation is mostly due to Poisson fluctuations in the number of stars in large volumes, and is an exceedingly slow process in galaxies. Hence in galactic dynamics we focus on fluctuations due to the motion of massive bodies (giant molecular clouds, spiral arms dark-halo lumps, star-clusters and satellite systems). Fluctuations cause stars to diffuse through action space, and this diffusion is observed in the solar neighbourhood. The diffusion coefficients can be calculated from the temporal power spectrum of the fluctuations or empirically determined from observations of solar-neighbourhood stars. Theory and observation are reasonably consistent, but there is plenty of scope for tightening constraints.
Spiral structure is an important source of fluctuations. Its dominant effect is the creation of transient resonances, which by first trapping and then releasing stars cause them to move from inside the corotation circle outwards, and vice versa. The random velocities of stars are not increased by such churning, but stars with similar ages but different metallicities are mixed up. In addition to churning the disc around corotation, spiral structure moves angular momentum outwards, from ILR to OLR, in the process heating the disc in the vicinity of the Lindblad resonances, especially the ILR.
Spiral structure is not fully understood because it is an inherently non-linear and global phenomenon. Lin-Shu-Kalnajs density-wave theory is restricted to the linear case and assumes tightly-would arms to make the physics essentially local. It predicts that tightly-wound leading waves propagate through a portion of the disc, unwinding as they go, so they inevitably violate the tight-winding approximation. As the waves pass from leading to trailing form, they are amplified by a process that lies beyond linear theory, and eventually their energy is thermalised at a Lindblad resonance by an analogue of Landau damping. The inaccuracy inherent in using the tight-winding approximation can be eliminated by solving the exact equation for normal modes. Such solutions confirm the basic picture derived with the tight-winding approximation but reveals a preference of spiral structure to lie inside CR. Unfortunately, van Kampen's work on electrostatic plasmas implies that the solutions of the normal-mode equation are not complete, so they do not provide a secure basis for understanding the dynamics of discs.
Self-gravitating discs are responsive systems because any disturbance is liable to excite leading waves, which may amplify significantly as they morph into trailing waves. Since the amplification becomes weaker as the disc heats, a pure stellar disc becomes less responsive as its velocity dispersion rises. Gas is an essential ingredient of a spiral galaxy because (i) it dissipates the energy of spiral waves, (ii) through star formation it constantly replenishes the population of stars with low velocity dispersion as spiral structure increases the velocity dispersion of older stars, and (iii) it makes any spiral gravitational potential observationally conspicuous by forming dust lanes and luminous blue stars near its troughs.
Galaxies live in the noisy environments of their dark halos, into which clumps with various masses are continually falling. As they pass through pericentre such lumps may excite spiral structure. Any spiral structure will quickly heat the disc. If the disc is relatively cold and therefore responsive, much more energy will be converted into heat than was imparted by the exciting lump: by shifting angular momentum out through the disc, spiral structure makes gravitational energy available for random motions, and thus increases the disc's entropy. It is important never to lose sight of the fact that a disc of stars on nearly circular orbits is occupying only a tiny fraction of the phase space that is energetically accessible to it. Any random process will scatter its stars into a broader distribution in phase space, and thus make it a hotter, thicker disc.