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Geometrická mechanika kontinua
Thesis title in Czech: Geometrická mechanika kontinua
Thesis title in English: Goemetric continuum mechanics
English key words: Lie algebra, Hamiltonian mechanics, Poisson bracket, fluid mechanics, solid mechanics, semidirect product
Academic year of topic announcement: 2018/2019
Type of assignment: diploma thesis
Thesis language:
Department: Mathematical Institute of Charles University (32-MUUK)
Supervisor: RNDr. Michal Pavelka, Ph.D.
Classical mechanics of point particles can be rewritten as Hamilton canonical equations, which are generated by a Hamiltonian and by the canonical Poisson bracket. This is the simplest case of Hamiltonian mechanics. In the case of continuum (or fields), for instance electromagnetism, kinetic theory, fluid mechanics, non-Newtonian fluid mechanics and solid mechanics can be also seen as being generated by a Poisson bracket and energy, and they are thus infinite-dimensional Hamiltonian systems. Hamiltonian mechanics is a geometrical approach unifying reversible evolution in continuum mechanics.

We shall study the following points:
1) Review the construction of fluid mechanics as a Lie-Poisson Hamiltonian system including entropy [1,2,3].
2) Construct the mechanics of distortion matrix (unifying the description of fluids, solids and non-Newtonian fluids) geometrically (semidirect product).[3]
3) Review the possibility of Hamiltonian numerical schemes for continuum mechanics [4].
[1] V.I. Arnold. Sur la géometrie différentielle des groupes de Lie de dimension infini et ses applications dans l’hydrodynamique des fluides parfaits. Annales de l’institut Fourier, 16(1):319–361, 1966.
[2] J Marsden, T Ratiu, and A Weinstein. Semidirect products and reduction in mechanics. Transactions of the american mathematical society, 281(1):147–177, 1984. ISSN 0002-9947. doi: 10.2307/1999527.
[3] Michal Pavelka, Václav Klika, and Miroslav Grmela. Multiscale Thermo-Dynamics. de Gruyter (Berlin), 2018.
[4] E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden, M. Desbrun, Geometric, Variational Discretization of Continuum Theories, Physica D: Nonlinear Phenomena (240(21), pp. 1724-1760, 2011)
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