A charged ellipsoidal bunch in a magnetic field

The states of a long rotating charged ellipsoidal bunch in a longitudinal uniform magnetic field are studied. The states are described using two integrals of motion that couple the transverse velocities (x)\dot]\dot x and (y)\dot]\dot y with the x and y coordinates; the frequency ω_H=eH/mc (where H is the total magnetic field); and the quantities ω₁ and ω₂, which characterize the Coulomb repulsion in the x and y directions. It is shown that equilibrium states with a high charge density per unit length (ν≳1) can exist.