Regular and chaotic motions in ion traps: A nonlinear analysis of trap equations

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Abstract
Using Lie group methods and prolongation techniques for vector fields, we present heretofore undiscovered symmetries of nonlinear-dynamical equations that describe ion motion in the Paul and in the Penning trap, respectively. The two-ion motion in the trap is investigated in detail. Checking integrability of the nonlinear equations, both Painlevé and modified Painlevé tests failed. However, using symmetry properties discovered by prolongation techniques, we are able to construct integrals of motion and to discover conditions for those parameter combinations that lead to regular motion in the trap. These parameters, which can be related to experimentally adjustable physical quantities, are listed. In the case of rotational invariance, the corresponding analytical solutions are derived from the integrals of motion. Parameter choices different from these values lead to chaotic motion, as demonstrated numerically by the calculation of Poincaré sections as well as by the determination of the Lyapunov spectrum.