Overview of Single-particle Nonlinear Dynamics

We give an overview of the single-particle non linear dynamics in circular accelerators. The main topics are: integration of equations of motion, fast symplectic tracking, dynamic aperture definition, long-term methods, quality factors and lattice optimization. Special emphasis is put on ideas and tools developed during the last decade

An Estimate of the Maximum Gradients in Superconducting Quadrupoles

In this paper we show that the electromagnetic design of several superconducting quadrupoles, built for particle accelerators, can be rather well approximated by a 36 degree sector coil with a wedge, canceling the first two field harmonics. We therefore carry out a complete analysis of this lay-out, obtaining an approximated equation for the critical gradient as a function of the coil area, magnet aperture, and of the superconducting properties of the cable. Using this model, we estimate through numerical methods the maximum critical gradient that can be obtained in quadrupole of a given aperture for Nb-Ti, Nb-Ti-Ta and Nb$_{3}$Sn

Normal Form Analysis of the LHC Dynamic Aperture

Normal form is a well developed tool to study the nonlinear dynamics of even the most complicated structures. This can become particularly useful in the design of large nonlinear hadron colliders like the LHC. In this study the correlation of various quality factors derived through normal forms with the dynamic aperture are used to investigate a machine that is dominated by octupolar errors

Long term stability in hadron colliders in presence of synchrotron oscillations and tune ripple

The problem of long-term losses in hadron colliders such as the Large Hadron Collider (LHC) is considered. A previous formula that provides the reduction of dynamic aperture with the number of turns is generalized to include also the relevant cases of off-momentum and tune ripple. The dynamic aperture turns out to shrink with a power of the inverse logarithm of the number of turns. Long-term tracking data of the LHC are analysed in this framework. The formula proves to hold in all cases, and the possibility of using its extrapolation to predict long-term losses are explored

Scaling Laws for Dynamic Aperture due to Chromatic Sextupoles

Scaling laws for the dynamic aperture due to chromatic sextupoles are investigated. The problem is addressed in a simplified lattice model containing 4 N identical cells and one linear betatron phase shifter to break the overall cell-lattice symmetry. Two families of chromatic sextupoles are used to compensate the natural chromaticity. Analytical formulae for the dynamic apertur as a function of the number of cells and of the cell length are found and confirmed through computer tracking

Dynamic aperture estimates and phase space distortions in nonlinear betatronic motion

Symplectic mappings which modelize the 4D betatronic motion in a magnetic lattice are considered. We define the dynamic aperture in terms of the connected volume in phase space of initial conditions which are bounded for a given number of iterations. Different methods for a fast estimate of this quantity are given; the analysis of the associated errors and the optimization of the integration steps are outlined. A comparison of the accuracy of these methods is given for both simple models and more realistic lattices

Numerical Evaluation of long-term Stability

The problem of predicting long-term particle loss in 4D betatronic motion is considered. A phenomenological scenario is derived through numerical tools based on tracking and frequency analysis. A three-parameter formula to interpolate the dynamic aperture versus the number of turns is proposed. The agreement with tracking data is excellent, and the extrapolation for very high number of turns agrees with the onset of chaos evaluated through the Lyapunov method

NERO: a code for evaluation of nonlinear resonances in 4D symplectic mappings

A code to evaluate the stability, the position and the width of nonlinear resonances in four-dimensional symplectic mappings is described. NERO is based on the computation of the resonant perturbative series through the use of Lie transformation implemented in the code ARES, and on the analysis of the resonant orbits of the interpolating Hamiltonian. The code is aimed at studying the nonlinear moti on of a charged particle moving in a circular accelerator under the influence of nonlinear forces