Integrable optics lens for the PSR and SNS

This is a study to increase the threshold of the Los Alamos LANSCE Proton Storage Ring (PSR) to the electron-proton (e-p) instability. The approach is to use a solenoid lens system to increase the Landau damping and betatron tune spread, without producing betatron resonances or energy spread that would necessitate increased aperture. This study rests upon the concept of integrable optics, which is a branch of Hamiltonian dynamics that determines how to modify dynamic systems to obtain integrable Hamiltonians having regular phase space maps and no chaos. For such cases, the particle trajectories, when integrated as functions of time and initial conditions, follow invariant surfaces. In essence, the maps are similar to linear maps with the addition of a betatron tune spread; however, the magnetic fields required to obtain such maps are complicated.