The interaction between geometric mechanics and numerical analysis is a rich source of insights and challenges for both the geometer and the analyst. The highly developed structures of conservative systems with symmetry provide both a powerful set of tools for the design of numerical methods for the integration of differential equations and an imposing array of criteria to be used in assessing the performance of those methods. Here we illustrate the interplay between some key components of the geometric theory of conservative systems on Lie groups and the design of numerical schemes for such systems by focusing on a single system with a high ratio of symmetry to degrees of freedom and using a combination of conservation laws, bundle trivializations, and matrix factorizations to develop several versions of the dynamics that are particularly well-suited for efficient numerical implementation.
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