= Solution
Put $S=\theta_1+\theta_2$ and $D=\theta_1-\theta_2$. The <gradient> has components $3\sin S+\sin D$ and $3\sin S-\sin D$, so a <stationary point> requires $\sin S=\sin D=0$. In the closed square, $S\in[0,2\pi]$ and $D\in[-\pi,\pi]$, and the compatible possibilities are precisely the four corners and $(\pi/2,\pi/2)$.
The <Hessian matrix> is
$$
\begin{pmatrix}3\cos S+\cos D&3\cos S-\cos D\\3\cos S-\cos D&3\cos S+\cos D\end{pmatrix}.
$$
Its <eigenvectors> $(1,1)$ and $(1,-1)$ have <eigenvalues> $6\cos S$ and $2\cos D$. The <second-derivative test> gives the following classification:
|| Point || $U$ || Hessian eigenvalues || Classification
| $(0,0),(\pi,\pi)$ | $-4$ | $6,2$ | <Minima>
| $(\pi,0),(0,\pi)$ | $4$ | $-6,-2$ | <Maxima>
| $(\pi/2,\pi/2)$ | $2$ | $-6,2$ | <Saddle point of a scalar function>
The corner classifications also hold for the smooth extension beyond the square, and hence for the restricted region. The global bounds $-4\leq U\leq4$ show that the stated <minima> and <maxima> are global within it. The central point has opposite curvature along the two diagonal directions.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1-dipole-energy-contours.png]
{title=Energy contours show two minima, two maxima and a central saddle}
{description=The $U=2$ contour crosses at the saddle. Filled markers locate all five points where the <gradient> vanishes; color and labeled <level curves> show how the energy changes between them.}
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