before, low energy dynamics are driven by a combination of the quadratic Zeeman energy, and the spin-dependent contact interaction. As we now show, in the case where $c_{2}$ is negative, these energies compete, driving a [*dynamic*]{} instability. Linearizing equation (\[eq:4\]) yields the relation: $$\label{eq:6}
1 = -\frac{4c^{2}_{2}\chi_{1}\chi_{2}}{(1+ (c_{0}+c_{2})\chi_{2})(1-(c_{0}+c_{2})\chi_{1})}$$ where the response functions are $\chi_{1} = \int \frac{d\vec{p}}{(2\pi)^{3}}\frac{(f_{0} + \frac{1}{2}\vec{k}\cdotp\vec{\nabla}_{p}f_{0})}{-\omega +\Omega_{d}+ \frac{\vec{k}\cdotp\vec{p}}{m}}$ and $\chi_{2} = \int \frac{d\vec{p}}{(2\pi)^{3}}\frac{(f_{0} - \frac{1}{2}\vec{k}\cdotp\vec{\nabla}_{p}f_{0})}{-\omega -\Omega_{d}+ \frac{\vec{k}\cdotp\vec{p}}{m}}$ and $\Omega_{d}  = (c_{0}-c_{2})n_{0}-q$. For $|k|v_{T} \ll c_{0}n_{0}$ and $\omega \ll c_{0}n_{0}$. The resulting dispersion is: $$\label{eq:7}
\omega^{2}(k) = q(q + 4c_{2}n_{0}) + (q + 2c_{2}n_{0})\frac{k^{2}v^{2}_{T}}{c_{0}n_{0}}$$

(50, 100) (-60, 3)[![\[fig:-2\] Microwave-field induced instability in a polar gas for $q<0$: The modes of a polar gas with ferromagnetic interactions ($c_{2} < 0 $) are plotted for $q = 4c_{2}n$ (top) and $q = 2c_{2}n$ (bottom) (hence $q < 0$). The gaps are given by $q(q + 2c_{2}n_{0})$ in each case. Note that for negative $q$, a finite wave-vector instability sets in. Increasing $|q|$ pushes the instability to larger wave-vectors. The dashed line is the most unstable frequency for $q=0$ (see Fig. \[fig:-1\](d)) given by $\omega^{2} = - (2c_{2}n_{0})^{2}$. The timescale for the onset of the instability is independent of $q$. We 