Geometric Mechanics & Symmetry: From Finite to Infinite Dimensions short course - day 4

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• Professor D.D.Holm, Imperial College London
• Thursday 30 May 2013, 15:30-17:30
• MR13.

If you have a question about this talk, please contact CCA.

Course notes available from http://www.maths.cam.ac.uk/postgrad/cca/files/darrylholmrevised.pdf

Classical mechanics, one of the oldest branches of science, has undergone a long evolution, developing hand in hand with many areas of mathematics, including calculus, differential geometry, and the theory of Lie groups and Lie algebras. The modern formations of Lagrangian and Hamiltonian mechanics, in the coordinate-free language of differential geometry, are elegant and general. They provide a unifying framework for many seemingly disparate physical systems, such as n-particle systems, rigid bodies and other continua, and electromagnetic and quantum systems.

This short course on Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to the geometric approach to classical mechanics, suitable for PhD students or advanced undergraduates. It fills a gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject. After a summary of the setting of mechanics using calculus on smooth manifolds and basic Lie group theory illustrated in matrix multiplication, the rest of the course considers how symmetry reduction of Hamilton’s principle allows one to derive and analyse the Euler-Poincaré equations for dynamics on Lie groups. The main topics are solitons in shallow water waves, ideal incompressible fluid dynamics and geophysical fluid dynamics (GFD).

Three worked examples that illustrate the course material in simpler settings are given full detail in the course notes. These will be assigned as outside reading and then discussed in Q&A sessions in class.

This talk is part of the Cambridge Centre for Analysis talks series.

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