  Our understanding of the mechanisms governing the structure and secular
evolution galaxies assume nearly integrable Hamiltonians with regular orbits;
our perturbation theories are founded on the averaging theorem for isolated
resonances. On the other hand, it is well-known that dynamical systems with
many degrees of freedom are irregular in all but special cases. The best
developed framework for studying the breakdown of regularity and the onset is
the Kolmogorov-Arnold-Moser (KAM) theory. Here, we use a numerical version of
the KAM procedure to construct regular orbits (tori) and locate irregular
orbits (broken tori). Irregular orbits are most often classified in
astronomical dynamics by their exponential divergence using Lyapunov exponents.
Although their computation is numerically challenging, the procedure is
straightforward and they are often used to estimate the measure of regularity.
The numerical KAM approach has several advantages: 1) it provides the
morphology of perturbed orbits; 2) its constructive nature allows the tori to
be used as basis for studying secular evolution; 3) for broken tori, clues to
the cause of the irregularity may be found by studying the largest, diverging
Fourier terms; and 4) it is more likely to detect weak chaos and orbits close
to bifurcation. Conversely, it is not a general technique and works most
cleanly for small perturbations. We develop a perturbation theory that includes
chaos by retaining an arbitrary number of interacting terms rather than
eliminating all but one using the averaging theorem. The companion papers show
that models with significant stochasticity seem to be the rule, not the
exception.
