= Charged circular elastic ring in a uniform magnetic field
{title2=$J=mr^2\dot\phi+qBr^2/2$}
A uniformly charged circular ring has
$$
L=\frac m2(\dot r^2+r^2\dot\phi^2)-2\pi^2k(r-r_0)^2+\frac{qBr^2}{2}\dot\phi.
$$
Here $k$ is the <spring constant> for extension of its full circumference and the <symmetric gauge> is used. The canonical <angular momentum> $J=mr^2\dot\phi+qBr^2/2$ is conserved. Eliminating the angle gives $m\ddot r=J^2/(mr^3)-q^2B^2r/(4m)-4\pi^2k(r-r_0)$. This model neglects electric self-interaction and noncircular deformation.
Back to article page
