Cubic polynomial for the motion of a heavy symmetric top

[Happiness]

In the second paragraph after the expression of $f(u)$ below, it wrote "there are three roots to a cubic equation and three combinations of solutions". However, the combination of having three equal real roots was not mentioned. Why?

In the next paragraph, in the second sentence, it wrote "at points $u=\pm1$, $f(u)$ is always negative, except for the unusual case where $u=\pm1$ is a root". But LHS of (5.62') $=\dot{u}^2=\dot{\theta}^2\sin^2\theta=0$ when $u=\cos\theta=\pm1$,i.e., $\theta=0$ or $\pi$. Thus RHS should also be 0. That means $u=\pm1$ is always a root to $f(u)=0$. What's wrong?

 

[gtoguy97]

The expression of $f(u)$ is given by:$$f(u)=Au^3+Bu^2+Cu+D$$The statement that there are three roots to a cubic equation and three combinations of solutions is referring to the fact that, for a cubic equation with real coefficients, there are three possible cases:1) all three roots are real and distinct 2) two of the three roots are equal and real 3) all three roots are equal and real. The statement is not mentioning the third case because it is already being handled in the next paragraph. The next paragraph is talking about the unusual case where one or both of the roots at $u=\pm 1$ is equal to 0. In this situation, the root at $u=\pm 1$ is not necessarily a real number, as the equation may have complex roots.