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Abstract

The stability of particle motion is a central concern of beam dynamics for the optimal design of particle accelerators. Linear effects are generally well under control, allowing us to store beams for more than 50 hours in machines like the Large Hadron Collider, at CERN. Non-linear effects, on the other hand, introduce complications that must be controlled to enhance the region of stable motion. In such complex systems, the emergence of chaos forces us to integrate the motion element-by-element to study the complete picture of beam dynamics. This establishes particle tracking as a fundamental tool required to probe non-linear effects. Yet despite the successes of the field, the ubiquity of the linear picture leads to a framework which often appears fragmented. In this respect, this dissertation proposes a formalism which can naturally describe linear, non-linear, analytic and numerical experiments alike using a self-consistent framework. To do so, we take an epicycle approach (in the spirit of the ancient Greeks) and place a general quasiperiodic expansion as the central object of study. Critically, this choice ensures close contact with empirical observables and enables a detailed spectral analysis of the particle’s motion. After providing a visual interpretation of KAM tori, we show how the integrals of motion can be recovered for arbitrary Hamiltonian flows, including fully coupled six-dimensional systems such as the LHC. Using Lie algebraic methods, we investigate the transport and deformation of these tori, establishing a foundation for describing coupled linear motion as a stepping stone toward a more complete non-linear treatment. Although closed-form solutions remain generally inaccessible, we demonstrate that the compensation of non-linear effects can be rigorously studied through the non-linear residual, a quantity shown to correlate strongly with dynamic aperture. Ultimately, the framework connects naturally with the Normal Form approach, which relies on the same quasiperiodic expansions.