Construction and Fourier analysis of invariant surfaces from tracking data

The authors study invariant surfaces in phase space by application of a symplectic tracking code. For motion in two degrees of freedom they use the code to compute I(s), Φ(s) for s=0, C, 2C, . . ., nC, where I =(I₁,I₂), Φ=(φ₁,φ₂) are action-angle coordinates of points on a single orbit, and C is the circumference of the reference orbit. As a test to see whether the orbit lies on an invariant surface (i.e. to test for regular and nonresonant motion), the authors fit the point to a smooth, piecewise polynomial surface I=I ˆ (φ₁,φ₂). The authors then compute additional points on the same orbit and test for their closeness to Iˆ. It is found that data from a few thousand turns are sufficient to construct accurate approximations to an invariant surface, even in cases with strong nonlinearities. Two-dimensional Fourier analysis of the surface leads to information on the strength of nonlinear resonances and provides the generator of a canonical transformation as a Fourier series in angle variables. The generator can be used in a program to derive rigorous bounds on the motion for a finite time