The Hamiltonian for a test particle is used to study the twohyphen;dimensional transverse particle dynamics in a nonhyphen;neutral charged particle beam propagating through a periodic quadrupole focusing field. The selfhyphen;electric and selfhyphen;magnetic fields produced by the beam space charge and current are taken into account. The Lie pointhyphen;symmetry group method and Noetherrsquo;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 Poincareacute; surfacehyphen;ofhyphen;section plots and Lyapunov exponents, have been performed to confirm the analytical results. A new numerical scheme, called thenumericalirreversibilitymethod, is proposed as an alternative approach to determine the integrability of Hamiltonian systems.
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