  The dynamical evolution of nearly half of the known extrasolar planets in
multiple-planet systems may be dominated by secular perturbations. The commonly
high eccentricities of the planetary orbits calls into question the utility of
the traditional Laplace-Lagrange (LL) secular theory in analyses of the motion.
We analytically generalize this theory to fourth-order in the eccentricities,
compare the result with the second-order theory and octupole-level theory, and
apply these theories to the likely secularly-dominated HD 12661, HD 168443, HD
38529 and Ups And multi-planet systems. The fourth-order scheme yields a
multiply-branched criterion for maintaining apsidal libration, and implies that
the apsidal rate of a small body is a function of its initial eccentricity,
dependencies which are absent from the traditional theory. Numerical results
indicate that the primary difference the second and fourth-order theories
reveal is an alteration in secular periodicities, and to a smaller extent
amplitudes of the planetary eccentricity variation. Comparison with numerical
integrations indicates that the improvement afforded by the fourth-order theory
over the second-order theory sometimes dwarfs the improvement needed to
reproduce the actual dynamical evolution. We conclude that LL secular theory,
to any order, generally represents a poor barometer for predicting secular
dynamics in extrasolar planetary systems, but does embody a useful tool for
extracting an accurate long-term dynamical description of systems with small
bodies and/or near-circular orbits.
