Example 5 In a triangle where the sum of two sides is $m$ and their included angle $\alpha$ is a constant, find the minimum value of the perimeter.

In a triangle where the sum of two sides is $m$ and their included angle $\alpha$ is a constant, find the minimum value of the perimeter.
Explanation: Let $m$, $\alpha$, and the two sides $x$, $y$ (variables) be the basic quantities, with the condition that $x+y=m$. Find the minimum value of the third side $c$.

Notice that
$$
\begin{array}{l}
c^{2}=x^{2}+y^{2}-2 x y \cos \alpha \\
=(x+y)^{2}-2 x y(1+\cos \alpha) \\
=m^{2}-2 x y(1+\cos \alpha) .
\end{array}
$$

Therefore, when $x=y=\frac{m}{2}$ (i.e., forming an isosceles triangle), $c^{2}$ has the minimum value $m^{2} \sin ^{2} \frac{\alpha}{2}$.
Thus, the minimum perimeter is $m+m \sin \frac{\alpha}{2}$.