Dynamics of secular evolution
  The text of lectures to the 2011 Tenerife Winter School. The School's theme
was "Secular Evolution of Galaxies" and my task was to present the underlying
stellar-dynamical theory. Other lecturers were speaking on the role of bars and
chemical evolution, so these topics are avoided here. We start with an account
of the connections between isolating integrals, quasiperiodicity and
angle-action variables - these variables played a unifying role throughout the
lectures. This leads on to the phenomenon of resonant trapping and how this can
lead to chaos in cuspy potentials and phase-space mixing in slowly evolving
potentials. Surfaces of section and frequency analysis are introduced as
diagnostics of phase-space structure. Real galactic potentials include a
fluctuating part that drives the system towards unattainable thermal
equilibrium. Two-body encounters are only one source of fluctuations, and all
fluctuations will drive similar evolution. We derive the orbit-averaged
Fokker-Planck equation and relations that hold between the second-order
diffusion coefficients and both the power spectrum of the fluctuations and the
first-order diffusion coefficients. From the observed heating of the solar
neighbourhood we show that the second-order diffusion coefficients must scale
as J^{1/2}. We show that periodic spiral structure shifts angular momentum
outwards, heating at the Lindblad resonances and mixing at corotation. The
equation that would yield the normal modes of a stellar disc is first derived
and then used to discuss the propagation of tightly-wound spiral waves. The
winding up of such waves is explains why cool stellar discs are responsive
systems that amplify ambient noise. An explanation is offered of why the
Lin-Shu-Kalnajs dispersion relation and even global normal-mode calculations
provide a very incomplete understanding of the dynamics of stellar discs.
