Nonlinear accelerator lattices with one and two analytic invariants

Abstract

Integrable systems appeared in physics long ago at the onset of classical dynamics with examples being Kepler's and other famous problems. Unfortunately, the majority of nonlinear problems turned out to be nonintegrable. In accelerator terms, any 2D nonlinear nonintegrable mapping produces chaotic motion and a complex network of stable and unstable resonances. Nevertheless, in the proximity of an integrable system the full volume of such a chaotic network is small. Thus, the integrable nonlinear motion in accelerators has the potential to introduce a large betatron tune spread to suppress instabilities and to mitigate the effects of space charge and magnetic field errors. To create such an accelerator lattice one has to find magnetic and electrtic field combinations leading to a stable integrable motion. This paper presents families of lattices with one invariant where bounded motion can be easily created in large volumes of the phase space. In addition, it presents 3 families of integrable nonlinear accelerator lattices, realizable with longitudinal-coordinate-dependent magnetic or electric fields with the stable nonlinear motion, which can be solved in terms of separable variables.

Advanced accelerator concept 2014 summary of WG7 muon colliders and advanced concepts

“…Theory predicts that lattices with one or two invariants and sufficiently strong nonlinear elements should suppress tune and envelope resonances without loss of stable phase space area [7]. Plans were presented to modify UMER to include nonlinear elements that meet the requirements for a singly invariant solution, which is well-suited to the current UMER framework.…”

“…Nonlinear integrable optics [7] is a promising route to large tune spreads to suppress parametric resonances such as beam halo. The Integrable Optics Test Accelerator (IOTA) has been proposed to study the practical implementation and beam dynamics of integrable lattices.…”

“…The concept of nonlinear integrable optics [7] shows promise for introducing order unity tune spread with large dynamic aperture, which can suppress space charge driven parametric resonances and associated beam loss via halo formation [8]. The Integrable Optics Test Accelerator (IOTA) is a novel ring designed [9] to explore these concepts, initially with low-charge electron bunches and subsequently with low-energy proton beams that include significant tune depression due to space charge.…”

Nonlinear dynamics with space-charge in a small electron recirculator

“…This tune spread can help mitigate resonances and halo formation through Landau damping, or nonlinear decoherence. We are undertaking a study for implementation of nonlinear focusing in UMER similar to the idea behind the IOTA project at Fermilab [9]. We plan to use a number of octupole magnets in a short straight section of the alternative lattice pictured in Fig.…”

Summary report of Working Group 7: Muon colliders and advanced concepts

“…There are at least two methods for creating an integrable lattice in practice [12]. One can use an electron lens, or one can make a nonlinear lens pro¿led to match the lattice beta function.…”