turkmath.org

The talk will be consisting of two main parts. In the first part, a gentle introduction to the geometric mechanics will be presented. Accordingly, some basic notions of the theory namely, Lie groups, Lie algebras, Poisson manifolds and Hamiltonian systems, Euler-Poincaré and Lie-Poisson equations, will be exhibited. Several examples will be provided. In the second part, we shall address the problem of determining the matched equations of motions of two interacting systems (whose configuration spaces are Lie groups) governing the coupled system starting with the individual equations of motions (in Lagrangian or/and Hamiltonian forms). The configuration spaces of the systems being Lie groups is imperative here in order to define the mutual actions. We shall present the theory of matched dynamics and particularly write the matched Lie-Poisson and the matched Euler-Poincaré equations. It will be shown that the theory of matched dynamics is a generalization of the well-developed semi-direct product theory. The theory will be discussed on a concrete example, the matched Lie-Poisson equations for the matched pair of upper and lower triangular matrix groups whose diagonal entries are 1.