Symplectic maps and chromatic optics in particle accelerators

Symplectic Tracking through Circular Accelerators with High Order Maps

It is discussed how high order transfer map8 generated using dift'erential algebraic methods can be used for symplectic tracking. Contrary to the usual tracking. the map approach makes it possible to study the specific properties of the system with as few approximations ..desired without prohibitive extra eft'ort. For example, the full Hamiltonian can be used. and the elementa can be treated wi...

Construction of symplectic maps for nonlinear motion of particles in accelerators.

Symplectic maps

Here Dfij = ∂fi/∂zj is the Jacobian matrix of f , J is the Poisson matrix, and I is the n × n identity matrix. Equivalently, Stokes’ theorem can be used to show that the loop action, A[γ] = ∮ γ pdq, is preserved by f for any contractible loop γ on X. If f preserves the loop action for all loops, even those that are not contractible, then it is exact symplectic. When n = 1, the symplectic condit...

Review of linear optics measurement and correction for charged particle accelerators

Measurement and correction of charged particle beam optics have been a major concern since the advent of strong focusing synchrotron accelerators. Traditionally, particle colliders have led the development of optics control based on turn-by-turn beam centroid measurements, while lepton storage rings have focused on closed-orbit-response matrix techniques. Recently, considerable efforts are bein...

Phantom Maps and Chromatic Phantom Maps

In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V ∧ X for V finite. In the second part, we introduce chromatic phantom maps. A map is n-phantom if it is null when restricted to finite spectra of type...