Many natural phenomena which occur in the realm of visual computing and computational physics, like the dynamics of cloth, fibers, fluids, and solids as well as collision scenarios are described by stiff Hamiltonian equations of motion, i.e. differential equations whose solution spectra simultaneously contain extremely high and low frequencies. This usually impedes the development of physically accurate and at the same time efficient integration algorithms. We present a straightforward computationally oriented introduction to advanced concepts from classical mechanics. We provide an easy to understand step-by-step introduction from variational principles over the Euler-Lagrange formalism and the Legendre transformation to Hamiltonian mechanics. Based on such solid theoretical foundations, we study the underlying geometric structure of Hamiltonian systems as well as their discrete counterparts in order to develop sophisticated structure preserving integration algorithms to efficiently perform high fidelity simulations.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors are grateful to Stefan Feess for preparing the simulation of the righting response of the turtle and its rendering. The reviewers' valuable comments that improved the manuscript are gratefully acknowledged. This work has been partially supported by the King Abdullah University of Science and Technology (KAUST baseline grants), the German Academic Exchange Service (Deutscher Akademischer Austauschdienst e.V.) funded by the government of the Federal Republic of Germany and the European Union, and the German National Merit Foundation (Studienstiftung des deutschen Volkes e.V.) funded by federal, state, and local authorities of the Federal Republic of Germany.