where $\textbf{1}$ is the identity matrix, Tr denotes the trace operation, and $\vec{\textbf{M}} = \text{Tr}(\vec{\textbf{S}}~\textbf{n})$ is the magnetization. Our form for the interaction energy is equivalent to that of Endo and Nikuni [@nikuni]. One may explicitly check that this interaction potential is rotationally invariant in spin space.

In experiments the external trapping potential $U$ is often spin-dependent, an effect which is readily incorporated. The spin independent interaction gives rise to self and cross interaction terms. The latter give rise to coherences and are encoded in the second term in (\[eq:3\]). In addition, the mean-field potential alters the trapping potential seen by all the atoms by an amount $c_{0}n_{tot}$, where the total density $n_{tot} = \text{Tr}(\textbf{n})$.

The contribution of the spin-dependent interaction is more subtle and can be understood as follows: The first term accounts for spin dynamics such as spin-relaxation collisions. For ferromagnetic interactions ($c_{2} < 0$), the last term increases the density of regions where the atomic spins are aligned with respect to one another ($ |M| = 1$). For a fully polarized gas in the $m_{F} = +1$ sublevel, this is $c_{2}\textbf{S}_{z}$, while for an unmagnetized ($m_{F} = 0$) gas it is zero.

Finally, we note that while 