Chaotic particle dynamics in a periodic focusing quadrupole magnetic field

The Hamiltonian for a test particle is used to study the two-dimensional transverse particle dynamics in a non-neutral charged particle beam propagating through a periodic quadrupole focusing field. The self-electric and self-magnetic fields produced by the beam space charge and current are taken into account. The Lie point-symmetry group method and Noether's theorem are applied to study the integrability of the equations of motion. It is concluded that the particle orbits are regular only for the case of a uniform density beam, and become chaotic when the beam density is nonuniform. Numerical computations, including Poincare surface-of-section plots and Lyapunov exponents, have been performed to confirm the analytical results. A new numerical scheme, called the numerical irreversibility method, is proposed as an alternative approach to determine the integrability of Hamiltonian systems