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Article

Устойчивые двумерные торы в ловушке Пеннинга при комбинированном частотном резонансе

We study the planar Penning traps in a resonance mode. The axial symmetry of the system is violated by deviation of the magnetic field from the trap axis at a small angle (the small parameter in the given model). The geometry of planar electrodes and their electric potentials are made consistent to reach a combined resonance, in both prime and subprime Hamiltonians under the small parameter expansion. In such a double-resonance regime we make the double averaging and derive the explicit formulas for the dependence of the averaged Hamiltonian on the controlling parameters of the trap. After the double reduction with respect to the primary and the secondary symmetry algebras, for the reduced Hamiltonian, an algorithm and explicit formulas for calculating all equilibrium points, explicit formulas for the energies and for the Hessians at these points in terms of the initial controlling parameters of the trap are obtained. In the original six-dimensional phase space the stable equilibrium points are related to invariant two-dimensional tori winded by near-periodic trajectories of a charge moving in the trap.