  We study the spin dynamics of quasi-one-dimensional F=1 condensates both at
zero and finite temperatures for arbitrary initial spin configurations. The
rich dynamical evolution exhibited by these non-linear systems is explained by
surprisingly simple principles: minimization of energy at zero temperature, and
maximization of entropy at high temperature. Our analytical results for the
homogeneous case are corroborated by numerical simulations for confined
condensates in a wide variety of initial conditions. These predictions compare
qualitatively well with recent experimental observations and can, therefore,
serve as a guidance for on-going experiments.
