Happy Birtday Q-Ball, Fix (at the same day as his brother, how strange :roll: ) and Sepap. Here's your present Qball : Spoiler E_{\omega} = E + \omega \left[ Q - \frac{1}{2i} \int d^{3} x(\phi^{*} \partial_{t} \phi - \phi \partial_{t} \phi^{*}) \right], where the energy is defined as E = \int d^{3} x \left[ \frac{1}{2} \dot{\phi}^{2} + \frac{1}{2} |\nabla \phi|^{2} + U(\phi, \phi^{*}) \right], and ω is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional Eω as E_{\omega} = \int d^{3} x \left[ \frac{1}{2} |\dot{\phi} - i \omega \phi|^{2} + \frac{1}{2} |\nabla \phi|^{2} + \hat{U}_{\omega}(\phi, \phi^{*}) \right] where \hat{U}_{\omega} = U - \frac{1}{2} \omega^{2} \phi^{2} . Since the first term in the functional is now positive, minimization of this terms implies \phi(\vec{r},t) = \phi_{0}(\vec{r}) e^{i\omega t}. We therefore interpret the Lagrange multiplier ω as the frequency of oscillation of the field within the Q-ball. (apparently there is no mathematic tool on the forum) Here I found a nice owl cake
why do i suddenly get the urge to kiss you guys! [youtube]http://www.youtube.com/watch?v=-4tCKuVaGKM[/youtube]