Geometric integrators in mechanics - the need for computer algebra tools
Résumé
In this contribution we will describe some objects of the generalized geometry that appear naturally in the qualitative analysis of mechanical systems. In particular we will discuss the Dirac structures within the framework of the systems with constraints as well as of the port-Hamiltonian systems. From the mathematical point of view, Dirac structures generalize simultaneously symplectic and Poisson structures. As for mechanics, the idea is to design numerical methods that preserve these structures and thus guarantee good physical behaviour in the simulation. Then, we will present a framework which is even more general-the one of differential graded manifolds (also called Q-manifolds), and discuss some possible ways of using them for the "structure preserving integrators" in mechanics. For all of the mentioned constructions we will explain the problems that arise in generic situations-most of them are open, but we think they are suitable for handling with various computer algebra approaches.
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