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Exact solutions with matter fields
Kokoška, David ; Ortaggio, Marcello (advisor) ; Žofka, Martin (referee)
In this thesis we investigate Robinson-Trautman solutions of Einstein's gravity cou- pled to a matter field in higher dimensions, specifically a conformally invariant and non- linear electromagnetic field. The latter possesses in general a non-zero energy-momentum tensor, which provides a source term in Einstein's equations. We focus concretely on an electromagnetic field aligned with the null vector field generating the expanding con- gruence of Robinson-Trautman spacetimes. At the beginning, we review the concept of optical scalars for a null vector field in higher dimensions and we use those to define the higher-dimensional Robinson-Trautman class of spacetimes. Next, we solve the corre- sponding Einstein's equations and present the complete family of exact solutions of the theory under consideration. We then contrast the obtained results with the known ones for the linear Maxwell theory in higher dimensions. As a check, we also compare our results to the well-known results in D = 4, since in this case our matter theory reduces to the standard linear Maxwell theory. Finally, we study properties of a subfamily of solutions which represent the static black holes within our class. In particular, we ana- lyze the asymptotic behaviour, we show that a curvature singularity is always present for r → 0 and the...
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Integrability in Hamiltonian machanics
Kokoška, David ; Krýsl, Svatopluk (advisor) ; Švarc, Robert (referee)
Title: Integrability in Hamiltonian mechanics Author: David Kokoška Department: Mathematical Institute of Charles University Supervisor: doc. RNDr. Svatopluk Krýsl, Ph.D., Mathematical Institute of Char- les University Abstract: Hamiltonian mechanics can be formulated using symplectic manifolds and so called Hamiltonian systems. In the Theorem of Liouville-Arnold, conditi- ons are described, under which solutions of Hamilton equations stay on a torus of dimension equal to the dimension of the configuration space. Examples on application of the Liouville-Arnold theorem are contained. We study the pro- blem of motion in a gravitational central force field in the connection with the Runge-Lenz vector. Keywords: symplectic manifold, hamiltonian system, Liouville-Arnold theorem, Kepler's problem 1
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