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From modelling of systems with constraints to generalized geometry and back to numerics
Author(s) -
Salnikov Vladimir,
Hamdouni Aziz
Publication year - 2019
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201800218
Subject(s) - double pendulum , context (archaeology) , integrator , dirac (video compression format) , kinematics , mathematics , mechanical system , pendulum , stability (learning theory) , dynamical systems theory , computer science , inverted pendulum , classical mechanics , nonlinear system , artificial intelligence , physics , paleontology , computer network , bandwidth (computing) , quantum mechanics , machine learning , nuclear physics , neutrino , biology
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‐type system 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.