Beams Document 4378-v1

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A Model Ring With Exactly Solvable Nonlinear Motion

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Fernanda G Garcia
Updated by:
Fernanda G Garcia
Document Created:
04 Jun 2013, 15:07
Contents Revised:
04 Jun 2013, 15:16
Metadata Revised:
04 Jun 2013, 15:16
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Recently, a concept of nonlinear accelerator lattices with two analytic invariants has been proposed. Based on further studies,the Integrable Optics Test Accelerator (IOTA) was designed and is being constructed at the Fermi National Accelerator Laboratory. Such a nonlinear lattice may be helpful in suppression of the collective instabilities by introducing a relatively large tune spread in a beam, while reducing phase-space area occupied by chaotic trajectories. Despite the clarity and transparency of the proposed idea, the detailed analysis of the beam motion remains quite complicated and should be understood better even for the case when no perturbations are taken into account. In this presentation we will review one of the three proposed realizations of the integrable optics, where the variables separation is possible in polar coordinates. This system allows for an exact analytical solution expressed in terms of elliptic integrals and Jacobi elliptic functions. It is worth noting that this model is valuable just because of this fact; the existence of the exact solution allows both, the possibility to check numerical algorithms used for tracking and to perform more rigorous analysis of the motion in comparison with the "crude" analysis of the topology of the phase space. In addition we will discuss some difficulties associated with numerical simulations of such a comparatively complex dynamical system (e.g. the use of high-order symplectic integrators) and will take a look at the possible perturbations for a model machine.
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