Example 3 Given $x+2 y+3 z+4 u+5 v=30$, find the minimum value of $w$ $=x^{2}+2 y^{2}+3 z^{2}+4 u^{2}+5 v^{2}$ (Example 6 from reference [1]).
By Cauchy-Schwarz inequality, we have $(1+2+3+4+5)\left(x^{2}+\right.$ $\left.2 y^{2}+3 z^{2}+4 u^{2}+5 v^{2}\right) \geqslant(x+2 y+3 z+4 u+5 v)^{2}$, i.e., $15 w \geqslant 30^{2}, \therefore$ the minimum value of $w$ is 60. (At this point, $x=y=z$ $=u=v=2)$