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 nonlinear 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 ongoing experiments. DOI: 10.1103/PhysRevLett.99.020404 PACS numbers: 05.30.ÿd, 03.75.Hh, 03.75.Kk, 03.75.Lm Spinor condensates realized by optically trapped ultracold atoms in a hyperfine Zeeman manifold allow us to address a broad scope of problems that are related to magnetic ordering. These include quantum phase transitions, exotic topological defects, and spin domains either within a mean-field regime or within the strongly correlated one [1]. Already in the mean-field regime the dynamics of spinor condensates shows some intriguing features, with an apparent randomness regarding the evolution from a given initial state toward a steady state, accompanied by formation of spin domains. Even for the simplest case of an F 1 spinor condensate, the interplay between spin-spin interactions, nonlinear terms, and temperature effects rends the analysis of the dynamics rather complex. Previous studies of the spinor dynamics show that, at early stages of the evolution, there is a coherent population transfer between the different hyperfine coupled sublevels [2,3]. Inclusion of temperature (T) not only smears out the population transfer [2], but also leads to a different distribution of population among the different hyperfine levels.In this contribution we analyze the dynamics of spinor F 1 condensates at both zero and finite T from a new perspective. As we shall show, the complex dynamics displayed by these systems can be understood in terms of oscillations in phase space around a steady state. The configuration of this state can be approximately determined by analyzing the trajectories of constant energy in the homogeneous case. To a good approximation, the populations that characterize this state are rather close to those that minimize the energy associated to spin-spin interactions for a given magnetization. In contrast, at finite T, the system is able to exchange energy with the thermal clouds. At large enough temperature, the populations of the steady state can be simply determined by maximizing the entropy of the homogeneous condensate. This leads to a different longtime configuration of the populations than the one attained starting from the same initial conditions at T 0. Our claims are supported by analytical results for a homogeneous condensate and by numerical investigations for trapped condensates. They provide a good qualitative agreement with recent experimental data on the dy...