Modes of a stellar system I: ergodic systems
  The excursions of star clusters and galaxies around statistical equilibria
are studied. For a stable ergodic model Antonov's Hermitian operator on
six-dimensional phase space has the normal modes as its eigenfunctions. The
excitation energy of the system is just the sum of the (positive) energies
associated with each normal mode. Formulae are given for the DFs of modes,
which are of the type first described by van Kampen rather than Landau, and
Landau `modes' can be expressed as sums of van Kampen modes. Each van Kampen
mode comprises the response of non-resonant stars to driving by the
gravitational field of stars on a group of resonant tori, so its structure is
sensitive to the degree of self gravity. The emergence of global distortions in
N-body models when particles are started from an analytical equilibrium is
explained in terms of the interplay of normal modes. The positivity of modal
energies opens the way to modelling the thermal properties of clusters in close
analogy with those of crystals.
