Betatron frequency and the Poincaré rotation number

Nagaitsev, Sergei; Zolkin, Timofey

doi:10.1103/physrevaccelbeams.23.054001

Cited by 8 publications

(19 citation statements)

References 24 publications

(23 reference statements)

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“…Since level set is polygon, each integral can be seen as a sum of integrals along its sides. If polygon has vertical sides, corresponding integrals should be replaced with (see [40,41])…”

Section: B Methods IImentioning

confidence: 99%

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Machine-assisted discovery of integrable symplectic mappings

Zolkin,

Kharkov,

Nagaitsev

2022

Preprint

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I. Introduction II. 2D symplectic mappings and integrability: historical overview III. 1-piece maps A. Polygon maps with integer coefficients B. Fundamental polygons of the first kind IV. 2-piece maps A. Linear mappings 1. Polygon maps with integer coefficients 2. Fundamental polygons of the second kind 3. Ingredients of mappings with polygon invariant B. Nonlinear mappings 1. Zoo mappings 2. Nonlinear integrable maps with polygon invariants V. Search algorithm VI. 3-piece maps A. Isolated d B. Discrete d, (D) C. Continuous d, (C) VII. 4-piece maps A. Isolated d and r B. Isolated d, discrete r C. Isolated d, continuous r D. Additional cases 1. Continuous and semi-continuous d with unity r. 2. Non-unity values of r. VIII. Applications of mappings with polygon invariants A. Connection to Painlevé equations and Suris mappings B. Near-integrable systems C. Discrete perturbation theory IX. Summary X. Acknowledgments A. Calculation of action-angle variables for polygon mappings 1. Action 2. Rotation number and angle variable a. Method I b. Method II B. Pseudo-code for the search algorithm C. Action-angle variables for 4-piece mappings D. Properties of invariant polygons for Zoo maps References

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“…Since level set is polygon, each integral can be seen as a sum of integrals along its sides. If polygon has vertical sides, corresponding integrals should be replaced with (see [40,41])…”

Section: B Methods IImentioning

confidence: 99%

“…Below we provide 2 methods of calculation of ν; both methods are based on Danilov theorem, see [40][41][42]. In addition, along with the second method we will show how to verify the integrability of the map.…”

Section: Rotation Number and Angle Variablementioning

confidence: 99%

Machine-assisted discovery of integrable symplectic mappings

Zolkin,

Kharkov,

Nagaitsev

2022

Preprint

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“…For exact expressions of tunes as functions of amplitudes, see Ref. [9]. A further generalization of Eq.…”

Section: Mcmillan Lensmentioning

confidence: 99%

McMillan electron lens in a system with space charge

Nagaitsev¹,

Lobach²,

Stern³

et al. 2021

J. Inst.

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Space charge (SC) forces of a circulating beam in a ring have both linear (defocusing) and nonlinear components, due to a nonuniform beam distribution. The linear component of SC forces produces a betatron tune shift, which is the largest for a zero-amplitude particle, while the nonlinear component produces an amplitude-dependent betatron tune spread. These SC effects are responsible for several undesirable phenomena in accelerators: emittance growth, particle losses, beam halo, etc. In this paper, we investigate the possibility to mitigate the SC forces by a thin McMillan lens providing an axially-symmetric kick, which is qualitatively opposite to the accumulated effect of beam's own SC. Experimentally, the proposed concept can be tested in Fermilab's IOTA ring. A thin McMillan lens can be implemented by a short (70 cm) insertion of an electron beam with a specifically chosen density distribution in transverse directions. In this article, to test if McMillan lenses can reduce the tune spread induced by SC, we make several simulations with a 6-D particle tracking code, Synergia. We choose such beam and lattice parameters that the SC tune spread is roughly 0.5 and the emittance growth due to the half-integer resonance is clearly observed without the SC compensation. Then, we try to reduce the emittance growth by adjusting the bare betatron tunes using the ring quadrupoles, and reducing the tune spread by the McMillan lenses. The results of reducing a large tune spread (≈ 0.5), reported here, are not perfect, but substantial. There is still room for further investigation. The simulations performed so far indicate that McMillan lenses can cope with a tune spread of ≈ 0.1 per lens.

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“…The tune distribution generated by the nonlinear McMillan map was studied analytically in ref. [20]. The theoretical treatment predicts the radial and angular Poincaré rotation numbers (i.e., the betatron tunes of a particle) from the two constants of motion.…”

Section: Analytical Expressions For the Detuningmentioning

confidence: 99%

“…A slightly different notation was used in ref. [20], based on the dimensionless variables = where K [ ] is the complete elliptic integral of the first kind and F [ , ] is the incomplete elliptic integral of the first kind. The variables and are defined as…”

Section: Analytical Expressions For the Detuningmentioning

confidence: 99%