Introduction:
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In the 1960s new, powerful techniques from modern differential geometry and topology were introduced into the study of dynamical systems. This new field, now called Geometric Mechanics, has successfully reformulated quantum and classical analytic mechanics in geometric language and brought in new methods from topology and analysis. Geometric Mechanics has experienced a spectacular growth in the last years impacting all adjacent mathematical fields as well as mathematical physics and certain areas of engineering, particularly control and robotics. The guiding idea in Geometric Mechanics is the application of techniques and methods of differential geometry and Lie group theory to the formulation and analysis of dynamical systems (classical, field theoretical, or quantum). Symplectic structures and their natural generalizations (the Poisson and Dirac manifolds) comprise the natural framework in the description and study of various phenomena that appear in classical (quantum) mechanics, among which we mention the following: symmetry reduction (both for finite and infinite dimensional systems) in a classical and quantum setting, Hamilton-Jacobi theory, mechanical systems that are subjected to external (possibly non-holonomic) constraints, the modeling of friction, geometric methods for numerical integration . . .
The aim of the session is to provide a forum to discuss recent significant research efforts in Geometric Mechanics that reveal the current challenges on this topic. |
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