Abstract
Linear coupling in a storage ring is conveniently analyzed in terms of transformations that put the single-turn map into block-diagonal form. Such a transformation allows us to define new variables, in which the dynamics are uncoupled. In this paper it is shown how a similar approach may be taken to nonlinear coupling, but that to decouple the map completely one needs to use a time-dependent canonical transformation. In Sec. III, we present a numerical example, based upon the analysis presented in previous sections, of a nonlinear transformation. In part for pedagogical reasons, and in part to make our use of notation clear, in Appendix A we reproduce the theory, along with a numerical example, of the well-known result for a linear transformation.
Highlights
Making transformations to reduce a dynamical system to as simple a form as possible is a familiar procedure in accelerator physics
Consider the phase space picture produced by a particle making multiple turns through a linear storage ring lattice
If the motion is symplectic and uncoupled, the values of the dynamical variables of the particle in any degree of freedom trace out smooth ellipses in phase space, with the area of each ellipse corresponding to a conserved quantity of the motion
Summary
Making transformations to reduce a dynamical system to as simple a form as possible is a familiar procedure in accelerator physics. If the motion is symplectic and uncoupled, the values of the dynamical variables of the particle in any degree of freedom trace out (over time) smooth ellipses in phase space, with the area of each ellipse corresponding to a conserved quantity of the motion. Well known is the normal form procedure for removing terms from nonlinear maps that drive resonances, including coupling resonances, and distort the phase space ellipses observed with uncoupled linear motion. Consider the case of a coupled system, in which the areas of the phase space ellipses in each degree of freedom correspond to conserved quantities, but where the coupling leads to the phase advance in one degree of freedom having a dependence on the amplitude of the motion in another degree of freedom. III that application of time-dependent decoupling transformations can be used to obtain information on the limits on the amplitudes for which the dynamics of a coupled system are stable
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