Abstract
The origin of nonlinear dynamics traces back to the study of the dynamics of planets with the seminal work of Poincaré at the end of the nineteenth century: Les Méthodes Nouvelles de la Mécanique Céleste, Vols.1-3 (Gauthier Villars, Paris, 1899). In his work he introduced a methodology fruitful for investigating the dynamical properties of complex systems, which led to the so-called "Poincaré surface of section," which allows one to capture the global dynamical properties of a system, characterized by fixed points and separatrices with respect to regular and chaotic motion. For two-dimensional phase space (one degree of freedom) this approach has been extremely useful and applied to particle accelerators for controlling their beam dynamics as of the second half of the twentieth century. We describe here an extension of the concept of 1D fixed points to fixed lines in two dimensions. These structures become the fundamental entities for characterizing the nonlinear motion in the four-dimensional phase space (two degrees of freedom).
Highlights
For experimental physicists, accelerators are devices that provide particle beams to a detector, or that guide two beams to collide with one another to study fundamental properties of matter at a subatomic level
We describe here an extension of the concept of 1D fixed points to fixed lines in two dimensions. These structures become the fundamental entities for characterizing the nonlinear motion in the four-dimensional phase space
It is less known that particles in a circular accelerator are subject to violent and complex dynamics, which in many aspects resemble and even exceed the complexity of the dynamics of planet motion around the sun
Summary
Week ending 12 JUNE 2015 particle interacting with islands, and extensive experimental and numerical studies have characterized the case of 1D dynamics [10,11]. The theory predicts the resonance stop band as a function of the driving term Despite this enormous progress, it remains unexplained how resonance structures in two or more degrees of freedom govern the phase space. The simplest situation of coupled nonlinear dynamics is found when considering a circular accelerator affected by a sequence of thin sextupoles. In this case, the equations of motion reads d2x ds þ kxðsÞx. Only two slowly varying harmonics remain while the others “average out” quickly and are ignored This approximation is valid close to the resonance, and for reasonable small nonlinear errors. In our study we find that Nj sextupolar errors lead to the slowly varying
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