Betatron frequency and the Poincare rotation number

Abstract

Symplectic maps are routinely used to describe single-particle dynamics in circular accelerators. In the case of a linear accelerator map, the rotation number (the betatron frequency) can be easily calculated from the map itself. In the case of a nonlinear map, the rotation number is normally obtained numerically, by iterating the map for given initial conditions, or through a normal form analysis, a type of a perturbation theory for maps. Integrable maps, a subclass of symplectic maps, allow for an analytic evaluation of their rotation numbers. In this paper we propose an analytic expression to determine the rotation number for integrable symplectic maps of the plane and present several examples, relevant to accelerators. These new results can be used to analyze the topology of the accelerator Hamiltonians as well as to serve as the starting point for a perturbation theory for maps.

References in corpus (5)

• IOTA (Integrable Optics Test Accelerator): Facility and Experimental Beam Physics Program
• Nonlinear Accelerator Lattices with One and Two Analytic Invariants
• Tevatron Electron Lenses: Design and Operation
• Detecting chaos in particle accelerators through the frequency map analysis method
• Accelerator-Feasible N-Body Nonlinear Integrable System

Cited by in corpus (9)

• Extracting Dynamical Frequencies from Invariants of Motion in Finite-Dimensional Nonlinear Integrable Systems
• Dynamics of McMillan mappings I. McMillan multipoles
• Integrable symplectic maps with a polygon tessellation
• Construction of approximate invariants for non-integrable Hamiltonian systems
• Dynamics of McMillan mappings II. Axially symmetric map
• Isochronous and period-doubling diagrams for symplectic maps of the plane
• Geometry of Almost-Conserved Quantities in Symplectic Maps. Part II: Recovery of approximate invariant
• Geometry of Almost-Conserved Quantities in Symplectic Maps. Part I: Perturbation Theory
• Geometry of Almost-Conserved Quantities in Symplectic Maps. Part III: Approximate Invariants in Nonlinear Accelerator Systems