Geometric mechanics examples simulation and theory

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Geometric Mechanics enhances the traditional approach to mechanics by the inclusion of ideas from geometry, nicely balanced with analytical methods. While this idea has its roots going back to the founders of mechanics, such as Jacobi, there has been a resurgence in the past few decades, with the infusion of many new ideas and links. For instance, it is a basic fact that the standard Hopf fibration, usually thought of as belonging to pure topology, already occurs in rigid body mechanics (going back to Euler and Lagrange). In another instance, one of the current important tools of Geometric Mechanics is certain coordinates, derived from symplectic geometry's various linear splittings together with the Darboux theorem. These sorts of features are natural in the modern setting of Geometric Mechanics. The symmetric, energy-conserving systems form an organizing center for the subject.  In these idealized models, friction is absent, multiple time scales are usually present, and the long time dynamics is very delicate.  Of considerable interest are solutions which are aligned with the symmetry group, which are called relative equilibria.  The physical form of these solutions depends on the symmetry, and can vary from circular motions, to screw motions and screw-spinning motions. Simulation of these systems is best done by numerical algorithms that preserve the systems' structures. Consequently, discrete (in time, or in space and time) versions of the systems, as well as structure preserving discretizations of the continuous systems, are an important ongoing research topic.  Control theory results in nonconservative systems, and is one of the main application areas of Geometric Mechanics. While the subject of mechanics has been established for centuries, the full differential geometric setting is relatively recent, and mathematically sophisticated. In Geometric Mechanics today, we find novel answers to dynamically relevant questions that more traditional approaches cannot ask, and we can systematically construct new numerical algorithms which have performance that traditional algorithms cannot approach.