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Local control based on chosen symmetries of carnot group
Frolík, Stanislav ; Galaev, Anton (referee) ; Zelenko, Igor (referee) ; Hrdina, Jaroslav (advisor)
Práce se zabývá analýzou geometrické teorie řízení na konkrétních Carnotových grupách. K ilustraci studovaného aparátu jsme zkontruovali novou třídu mechanismů, které jsme nazvali robotický červ. Tyto mechanismy přináší novou třídu kontrolních problémů neholonomní mechaniky. V práci budeme analyzovat konkrétní mechanismus, dvoučlánkového robotického červa. Můžeme uvažovat, že řízení tohoto mechanismu je dáno Carnotovou grupou s filtrací (4,6). K řešení optimálního problému vytvoříme nový návrh řízení. Naše metoda je založena na volbě dodatečné geometrické struktury na algebře řízení, hlavně využíváme CR strukturu. Pak můžeme volit počáteční třídu geodetik, kterou rozšíříme na celý konfigurační prostor.
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Order and chaos in a simple model of molecular vibrations
Novotný, Jakub ; Stránský, Pavel (advisor) ; Kloc, Michal (referee)
In this work we join the struggle to find a connection between classical and quantum chaos. We study the classical chaos in the classical limit of a simple algebraic model of molecular vibrations, based on the dynamical symmetry of u(3) algebra. We employ several methods for the classical chaos analysis, namely the Lyapunov exponents, the Poincaré sections, and the chaotic fraction of the classical phase space. It is shown that the chaoticity of the model depends nontrivially both on the strength of the external field and on the energy of the system. This work prepares the ground for a future comparison of classical and quantum chaos indicators. 1
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Symmetries of systems in spaces related to high-dimensional black hole spacetime
Kolář, Ivan ; Krtouš, Pavel (advisor) ; Kubizňák, David (referee)
In this work we study properties of the higher-dimensional generally rotating black hole space-time so-called Kerr-NUT-(A)dS and the related spaces with the same explicit and hidden symetries as the Kerr-NUT-(A)dS spacetime. First, we search commuta- tivity conditions for classical (charged) observables and their operator analogues, then we investigate a fulfilment of these conditions in the metioned spaces. We calculate the curvature of these spaces and solve the charged Hamilton-Jacobi and Klein-Gordon equations by the separation of the variables for an electromagnetic field, which pre- serves integrability of motion of a charged particle and mutual commutativity of the corresponding operators.
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Fyzika z teorie pravděpodobnosti
Gottvald, Aleš
Following basic ideas of information physics, probability theory features as the inner symmetries of physical laws. Consequently, we conjecture that many fundamental physical facts are already hidden in the unique logical structure of probability theory and need not be postulated. A link with statistical thermodynamics emerges via the exponential (MaxEnt) mapping between probability and entropy, whose scaling symmetry also makes a natural bridge to fractal physics and projective geometries. To facilitate links with many other symmetries and physical areas, the exponential mapping between Lie groups and Lie algebras is suggested as a generalization of the MaxEnt relationship. We point out that the natural space of probability theory is an intrinsically 6-dimensional manifold with two fundamental governing equations imposed, which gives a novel straightforward rationale for the emergence of the 4+6=10-dimensional hyperspace, particularly important in modern particle physics.
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