# Storage Rings

Storage rings of the charged particles are the backbone of the high energy circular colliders such as the Large Hadron Collider (LHC), where the Higgs particle was discovered in Geneva, Switzerland. They are also essential parts of the modern synchrotron light sources for research of material and medical sciences. They have many other applications with circumference ranging from a few meters to 27 kilometers.

The linear motion of the particles in the modern storage rings was well understood by Courant and Snyder in the theory of the alternating-gradient synchrotron. But the particle motion becomes nonlinear because of the sextuples introduced for chromatic compensations. In general, the nonlinearity in a periodic system like a storage ring generates nonlinear resonances, which are shown in the tune plane in the following figure and play an important role in the single-particle dynamics. The resonances define the topology and geometry of the phase space. Most importantly, the overlapping resonances lead to chaotic motion.

The canonical perturbation theory has been widely used in the particle accelerators. It has been successfully applied to the analysis of an isolated resonance. However, it fails to describe all nonlinear resonances in general due to the so-called problem of the small denominators resulting in the divergence of the perturbation series. In particular, the formal symplectic transformation to the Birkhoff normal form has to be divergent; otherwise the system is integrable in general.

This issue is only partially resolved by the well-known Kolmogorov-Arnold-Moser (KAM) theorem, moving away from the general solutions and focusing on the special ones, namely invariant tori. It shows that the non-resonance tori in an integrable system will be distorted but preserved, provided that the perturbation is sufficiently small. These survived tori provide definitive boundary to general orbit with a constant energy in a two-dimensional system and therefore ensure the orbit stability. For a three-dimensional system, the tori may not provide the stability of the orbit because of Arnold's diffusion. Given the sophisticated nature of the mathematics, a huge gap still exists between the theory and the practical design of circular accelerators.

Recently, we try to bridge the gap by studying simple periodical systems such as alternating gradient focusing cell and theoretical minimum emittance cell with general parameterization. These systems are chosen to be a good approximation of storage rings and yet simple enough to be studied analytically. Based on symplectic maps and Lie Algebra, we have gained many insights in the instability of the charged particles in the periodical systems. Our research will continue to more realistic cells such as double-bend and multi-bend achromat