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Model-independent analysis of beam dynamics in circular accelerators reveals spatial-temporal modes due to beam oscillations. Although such modes have been shown to be informative, their usefulness is limited by the lack of quantitative understanding in general. Here we present a simple approach to studying the modes analytically, which results in a quantitative understanding of the modes and an algorithm to extract the physical modes by untangling the mixed singular value decomposition modes. Particularly, we focus on the coupled betatron modes that are of great interest for high-energy colliders. A simple relationship between the coupling modes and the lattice functions is established, which not only provides a quantitative understanding but also lays the foundation for using these modes to measure machine properties and beam motion in phase space

Untangling mixed modes in model-independent analysis of beam dynamics in circular accelerators