Dinamik sistemlerin eşlenmesi

Karşılıklı etki-tepki içindeki iki fiziksel sistemin ortak hareketlerini veren denklemler (eşlenmiş Lie-Poisson ve eşlenmiş Euler-Poincaré) elde edilecektir. Eşlenmiş denklemlerin literatürde çokça çalışılmış yarı-direkt çarpım teorisinin genişlemesi olduğu gösterilecektir. İki örnek verilecektir. İlki, köşegen elemanları 1 olan alt ve üst üçgensel matris gruplarının oluşturduğu eşlenmiş Lie grubu üzerinde eşlenmiş Lie-Poisson denklemlerinin yazılmasıdır. İkinci örnek ise ikinci sınıf nilpotent grupların kendiyle eşlenmesi ile elde edilecek Lie grupları üzerinde eşlenmiş hareket denklemlerinin yazılmasıdır. İki yeni açık problem sunulacaktır. Bunlardan ilki, plazma ve akışkan arasında pür geometrik yapının eşlenmiş dinamik düzleminde ele alınması, diğeri ise karşılıklı etki-tepki içindeki iki kesikli sistemin eşlenmesidir.

Matching of dynamical systems

The equations (matched Lie-Poisson and matched Euler-Poincaré) are written for a couple of mutually interacting physical systems. It is shown that the matched dynamics is a generalization of the well-developed semi-direct product theory. Two examples are provided. The first one is to write the matched equations for the matched pair of upper and lower triangular matrix groups whose diagonal entries are 1. The second example is to write the matched equations for the Lie group obtained by matching a nilpotent group of class two by itself. Two new open problems are presented. One of these is to write pure geometric relation between the plasma and fluid in the framework of the matched dynamics. The other is to match two discrete systems under mutual interaction.

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