Example 2 If $a>0, b>0, c>0$, compare $a^{3}+b^{3}+c^{3}$ and $a^{2} b+b^{2} c+c^{2} a$.
Analysis: Although the problem does not specify the size relationship of the three numbers, the expressions being compared have symmetry. Therefore, we can assume the order of the three numbers without loss of generality.

Solution: Without loss of generality, let $a \geq b \geq c>0$, then $a^{2} \geq b^{2} \geq c^{2}>0$, thus $a^{3}+b^{3}+c^{3}$ is a sum in order, and $a^{2} b+b^{2} c+c^{2} a$ is a sum out of order. By the rearrangement inequality, we have $a^{3}+b^{3}+c^{3} \geq a^{2} b+b^{2} c+c^{2} a$.