Let $\chi$ be the angle between $\widehat{\mathbf p}$ and $\widehat{\mathbf n}$. Then
$$
p_iQ_{ij}p_j
=\lambda p^2(2\cos^2\chi-1)
=\lambda p^2\cos2\chi.
$$
For $\zeta,\lambda>0$, the coupling $-\zeta\lambda p^2\cos2\chi/2$ is minimized by $\cos2\chi=1$, so
$$
\boxed{\widehat{\mathbf p}=\pm\widehat{\mathbf n}}.
$$
The uniform terms depending on $p$ reduce to
$$
\frac12(a-\zeta\lambda)p^2+\frac b4p^4.
$$
Their nonzero minimum is
$$
\boxed{p^2=-\frac{a-\zeta\lambda}{b}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 2 / g / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-344/2/g/solution