Integrable Accelerator Lattices With Periodic And Exponential Invariants

This paper presents a new variety of one-dimensional nonlinear integrable accelerator lattices with periodic and exponential invariants in coordinates and momenta. Extension to two-dimensional transverse motion, based on a recently published approach, is discussed. The integrable accelerator lattices represent a continuation of linear systems with Courant-Snyder invariants to the nonlinear domain, where the frequencies of betatron motion 'strongly' depend on betatron amplitudes (the word 'strongly' means that the spread of betatron tunes is comparable to the tune itself). This spread can help to advance beam intensities by introducing a very large Landau damping. Recently, a possible method to realize stable integrable motion in accelerators with 2D transverse magnetic field was suggested (1). In principle, all 1D integrable lattices with short nonlinear lenses can be converted to 2D integrable lattices (we'll show examples of this conversion later in this paper). Reference (2) presented a method to find a vast variety of 1D and 2D integrable systems with invariants, polynomial in coordinates and momenta. The same method was used to find invariants that are harmonic or exponential functions of coordinates and momenta. Here we briefly present the theory and the method, along with solutions for lattices having nonlinear kicks with the aforementioned invariants, and show the behaviour of these integrable lattices in the 2D case with transverse magnetic fields.