Title: From modelling of systems with constraints to generalized geometry and back to numerics,
Abstract: In this note we describe how some objects from generalized geometry appear in
the qualitative analysis and numerical simulation of mechanical systems. In
particular we discuss double vector bundles and Dirac structures. It turns out
that those objects can be naturally associated to systems with constraints --
we recall the mathematical construction in the context of so called implicit
Lagrangian systems. We explain how they can be used to produce new numerical
methods, that we call Dirac integrators.
On a test example of a simple pendulum in a gravity field we compare the
Dirac integrators with classical explicit and implicit methods, we pay special
attention to conservation of constrains. Then, on a more advanced example of
the Ziegler column we show that the choice of numerical methods can indeed
affect the conclusions of qualitative analysis of the dynamics of mechanical
systems. We also tell why we think that Dirac integrators are appropriate for
this kind of systems by explaining the relation with the notions of geometric
degree of non-conservativity and kinematic structural stability.