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Abstract
The advent of circular accelerators based on superconducting magnets has revolutionized the field of beam dynamics, with particle motion turning from linear to nonlinear due to unavoidable high-order field errors generated by the ring magnets. Nonlinear dynamics was already well mastered, e.g., in the close field of celestial mechanics as similar problems had been considered and successfully tackled. Hence, several results were available to aid comprehension of the behavior of charged particle beams under the influence of nonlinear forces. Here, we discuss how concepts derived from the theory of dynamical systems, linked with the fundamental Kolmogorov--Arnold--Moser theory and Nekhoroshev theorem, can be successfully applied to the analysis of nonlinear motion of charged particles in a circular accelerator. Based on these ideas, an innovative method to measure the extent of the phase-space region within which bounded motion occurs is presented, which has been successfully tested for the first time at the CERN LHC.
Highlights
Dynamic aperture (DA) is the amplitude of the phase space region where stable motion occurs
The advent of circular accelerators based on superconducting magnets has revolutionized the field of beam dynamics, with particle motion turning from linear to nonlinear due to unavoidable high-order field errors generated by the ring magnets
Ideas from the theory of dynamical systems have been used to analyze the problem of nonlinear motion of protons in circular accelerators such as the Large Hadron Collider (LHC)
Summary
Dynamic aperture (DA) is the amplitude of the phase space region where stable motion occurs. Given the choice of the coordinates, DðNÞ is expressed in units of beam sigma In this way, dynamic aperture can be considered a function of time, with an asymptotic value representing the region of stability for arbitrary time. Computation consists of simulating the evolution of a large number of initial conditions, distributed to provide good coverage of the phase space under study, probing whether motion remains bounded over the time interval selected for the simulations. An example of results from such a simulation is shown in Fig. 1 (left), which represents a set of initial conditions in the polar grid of normalized physical space, for one configuration of the LHC machine at top-energy for the clockwise beam (the socalled Beam 1).
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