By Percy Deift

The idea of classical $R$-matrices presents a unified strategy to the certainty of such a lot, if no longer all, recognized integrable platforms. This paintings, that is appropriate as a graduate textbook in the fashionable thought of integrable structures, offers an exposition of $R$-matrix thought via examples, a few outdated, a few new. specifically, the authors build non-stop types of a number of discrete structures of the sort brought lately by way of Moser and Vesclov. within the framework the authors identify, those discrete platforms seem as time-one maps of integrable Hamiltonian flows on co-adjoint orbits of applicable loop teams, that are in flip comprised of extra primitive loop teams through classical $R$-matrix concept. Examples comprise the discrete Euler-Arnold most sensible and the billiard ball challenge in an elliptical zone in $n$ dimensions. past result of Moser on rank 2 extensions of a hard and fast matrix will be included into this framework, which means particularly that many famous integrable systems---such because the Neumann process, periodic Toda, geodesic stream on an ellipsoid, etc.---can even be analyzed through this technique.

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