scholarly journals Optimization of accelerator parameters using normal form methods on high-order transfer maps

2007 ◽  
Author(s):  
Pavel Snopok
Keyword(s):  
Normal Form ◽  
High Order ◽  
Transfer Maps ◽  
Normal Form Methods

scholarly journals ON THE REPRESENTATION OF MAPS BY LIE TRASFORMS

2012 ◽  
Author(s):  
Antonio Giorgilli
Keyword(s):  
Normal Form ◽  
Arbitrary Dimension ◽  
Lie Series ◽  
Straightforward Manner ◽  
Standard Map ◽  
Lie Transform ◽  
Normal Form Methods

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Schröder–Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension.

scholarly journals Computation and consequences of high order amplitude- and parameter-dependent tune shifts in storage rings for high precision measurements

2019 ◽  
Vol 34 (36) ◽  
pp. 1942011 ◽  
Author(s):  
Adrian Weisskopf ◽  
David Tarazona ◽  
Martin Berz
Keyword(s):  
Phase Space ◽  
Normal Form ◽  
High Precision ◽  
Storage Ring ◽  
Differential Algebra ◽  
Storage Rings ◽  
High Order ◽  
Precision Measurements ◽  
System Parameters ◽  
Parameter Dependent

Nonlinear effects of the various electric field and magnetic field components of storage rings to confine the particles and bend their trajectory can cause substantial amplitude-dependent tune shifts within the beam. Furthermore, tune shifts are often sensitive to variations of system parameters, e.g. total particle momentum offsets [Formula: see text]. Such amplitude- and parameter-dependent tune shifts influence the dynamics and stability of a beam in particle storage rings. Thus, it is critical for high precision measurements to analyze and understand these influences. On this basis, we present normal form methods for the calculation of high order amplitude and system parameter dependencies of the horizontal and vertical tunes in storage rings using the differential algebra (DA) framework within COSY INFINITY. A storage ring is simulated using COSY INFINITY to generate a DA Poincaré return map describing the transverse phase space behavior after each revolution in the storage ring. The map is expanded around the parameter-dependent closed orbit of the system before transforming the resulting map into normal form coordinates to extract the high order tune dependencies on the phase space amplitude and variation in the system parameters. As a specific example, a storage ring similar to the Storage Ring of the Muon [Formula: see text]-2 Experiment at Fermilab (E989) is investigated.

HIGH-ORDER DESCRIPTION OF THE DYNAMICS IN FFAGs AND RELATED ACCELERATORS

2009 ◽  
Vol 24 (05) ◽  
pp. 908-922 ◽  
Author(s):  
KYOKO MAKINO ◽  
MARTIN BERZ ◽  
PAVEL SNOPOK ◽  
CAROL JOHNSTONE
Keyword(s):  
Optimization Methods ◽  
Precise Determination ◽  
High Order ◽  
Minimal Impact ◽  
Closed Orbits ◽  
Dynamic Aperture ◽  
Resonance Strengths ◽  
Transfer Map ◽  
Normal Form Methods ◽  
Energy Dependent

In this paper, we describe newly developed tools for the study and analysis of the dynamics in FFAG accelerators based on transfer map methods unique to the code COSY INFINITY. With these new tools, closed orbits, transverse amplitude dependencies and dynamic aperture are determined inclusive of full nonlinear fields and kinematics to arbitrary order. The dynamics are studied at discrete energies, via a high-order energy-dependent transfer map. The order-dependent convergence in the calculated maps allows precise determination of dynamic aperture and detailed particle dynamics. Using normal form methods, and minimal impact symplectic tracking, amplitude- and energy-dependent tune shifts and resonance strengths are extracted. Optimization by constrained global optimization methods further refine and promote robust machine attributes. Various methods of describing the fields will be presented, including representation of fields in radius-dependent Fourier modes, which include complex magnet edge contours and superimposed fringe fields, as well as the capability to interject calculated or measured field data from a magnet design code or actual components, respectively.

scholarly journals High order transfer maps for perturbed Keplerian motion

2015 ◽  
Vol 122 (4) ◽  
pp. 333-358 ◽  
Author(s):  
Alexander Wittig ◽  
Roberto Armellin
Keyword(s):  
High Order ◽  
Keplerian Motion ◽  
Transfer Maps
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