A rotating frame of reference (since you rotate your BEC to generate these), where the energy is given by:

$\stackrel{~}{E}[\mathrm{\Psi}]=E[\mathrm{\Psi}]-L[\mathrm{\Psi}]\cdot \mathrm{\Omega},$

Where $\mathrm{\Omega}$ is the rotational velocity which you apply to the BEC.

Now the argument that the term $L[\mathrm{\Psi}]\cdot \mathrm{\Omega}$ should be substracted comes from the fact that in a rotating frame your system loses that fraction of rotational energy. Now I was wondering if anyone knew where the form of $L[\mathrm{\Psi}]\cdot \mathrm{\Omega}$ came from?

I know that in a rotating frame of reference you have that $\overrightarrow{v}={\overrightarrow{v}}_{r}+\overrightarrow{\mathrm{\Omega}}\times \overrightarrow{r}$ . If you fill this in into the kinetic energy and use some basic definitions, you get that the extra effect of rotation is given by:

$\frac{1}{2}I{\mathrm{\Omega}}^{2}=\frac{1}{2}J\mathrm{\Omega}.$

This yields half of the value that is used in the book, is there something that I'm missing or not seeing right ?