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Understanding the locomotion of animals and robots can be a challenging problem, involving nonlinear dynamics and the coordination of many degrees of freedom. Differential geometry and gauge theory offer a vocabulary for discussing these dynamics in terms of lengths, areas, and curvatures. In particular, a tool called the *Lie bracket* combines these geometric concepts to describe the effects of cyclic changes in the locomotor's shape, such as the gaits used by walking or crawling systems.
In this talk, I will review some basic principles of differential geometry and Lie group theory, and show how they provide insight into the locomotion of undulating systems (such as snakes and micro-organisms). I will then discuss my work on how coordinate representations affect the information provided by the geometric structures, and show that the choice of coordinates for a given system can be optimized in a simple, fundamental manner. Building on these results, I will demonstrate that the geometric techniques are useful beyond the "clean" ideal systems on which they have traditionally been developed, and can provide insight into the motion of systems with considerably more complex dynamics, such as locomotors in granular media.