Numerical evaluation of the integrals of motion in particle accelerator tracking codes

Particle tracking codes are one of the fundamental tools used in the design and the study of complex magnetic lattices in accelerator physics. For most practical applications, non-linear lenses are included and the Courant-Snyder formalism falls short of a complete description of the motion. Likewise, when the longitudinal motion is added, synchro-betatron coupling complicates the dynamics and different formalisms are typically needed to explain the motion. In this paper, a revised formalism is proposed based on the Fourier expansion of the trajectory -- known to be foundational in the KAM theorem -- which naturally describes non-linear motion in 2D, 4D and 6D. After extracting the fundamental frequencies and the Fourier coefficients from tracking data, it is shown that an approximate energy manifold (an invariant torus) can be constructed from the single-particle motion. This cornerstone allows to visualize and compute the areas of the torus projections in all conjugate planes, conserved under symplectic transformations. These are the integrals of motion, ultimately expressed in terms of the Fourier coefficients. As a numerical demonstration of this formalism, the case of the 6-dimensional Large Hadron Collider (LHC) is studied. Examples from the 2D and 4D H\'enon map are also provided. Even for heavily smeared and intricate non-linear motion, it is shown that invariant tori accurately describe the motion of single particles for a large region of the phase space, as suggested by the KAM theorem.