Example 3 Find the minimum value of the function with real variables $x$ and $y$
$$
u(x, y)=x^{2}+\frac{81}{x^{2}}-2 x y+\frac{18}{x} \sqrt{2-y^{2}}
$$

(2nd Hope Cup for High School Grade 2)
Solution: Completing the square, we get
$$
u=(x-y)^{2}+\left(\frac{9}{x}+\sqrt{2-y^{2}}\right)^{2}-2 \text {. }
$$

Consider points $P_{1}\left(x, \frac{9}{x}\right)$, $P_{2}\left(y,-\sqrt{2-y^{2}}\right)$ on the plane. When $x \in \mathbf{R}, x \neq 0$, the trajectory of $P_{1}$ is a hyperbola with the two coordinate axes as asymptotes. When $y \in \mathbf{R},|y| \leqslant \sqrt{2}$, the trajectory of $P_{2}$ is the lower half of a circle centered at the origin with a radius of $\sqrt{2}$. Geometrically, it is evident that the minimum value of $\left|P_{1} P_{2}\right|$ is $3 \sqrt{2}-\sqrt{2}$, from which we can derive
$$
u_{\text {min }}=(3 \sqrt{2}-\sqrt{2})^{2}-2=6 \text {. }
$$

Note: A slight variation of this example can yield the 1983 Putnam Mathematical Competition problem for college students:

Find the minimum value of $f(u, v)=(u-v)^{2}+\left(\sqrt{2-u}-\frac{9}{v}\right)^{2}$.