Statistical stability of three and more body hierarchical systems in celestial mechanics

itqquad Then to the Heav'n itself I cried,qquad Asking, `What Lamp had destiny to guideqquad Her little Children stumbling in the Dark?'qquad And - `A blind understandingMissing data. Heav'n replied. qquadqquadqquad Rubaiyat of Omar Khayyam It seems that not everybody in Persia in the eleventh century was as convinced as the astrologers that the movements of the heavens controlled the destiny of Man. Nevertheless, for many centuries before and since, kings and emperors rewarded handsomely those astronomer/astrologers who could give them advice based on the movements of the planets and other celestial bodies. (There may be some astronomers today who would wish for similar generous patronage). Since the advent of modern celestial mechanics with the work of Isaac Newton, orbital motion has been studied for its own sake, and in the last thirty years, for the purposes of sending artificial satellites and manned craft into space. Yet for 300 years, one of the most important questions posed by celestial mechanics remains unanswered: are the motions of the planets in the Solar System stable? Could planets collide or even escape? Countless workers since Newton's time have sought Lamps to the destiny of the Solar System, but our Understanding is still obscured by many blind-spots. This thesis does not claim to give any definitive answers to these questions. It does indicate how to obtain quantitative estimates of the likelihood of certain events occurring. Simple statistical methods are applied to the results of numerical experiments and give probabilities of planetary orbits crossing or bodies escaping dynamical systems altogether. In Chapter 1 a general review of the problem of the Solar System's stability is given along with brief descriptions of methods and definitions of stability which have been used in the past. This thesis studies the stability of real and fictitious dynamical systems not necessarily associated with the Solar System. It investigates one particular definition of stability, namely hierarchical stability, using special perturbation methods. The definitions of hierarchical systems, hierarchical stability and empirical stability parameters are reviewed in Chapter 2. These will form the basis for subsequent numerical experiments. One further definition of stability - Hill stability is an important condition for hierarchical stability. It has been studied in a mathematically rigorous way in the problem of three massive bodies in mutually perturbed orbits. This analysis as well as some new numerical results are given in Chapter 3. Numerical integration experiments were carried out, with the aid of a mainframe computer, to study the period of time for which various three-body systems remain stable. Several hundred fictitious systems with different masses and starting conditions were studied. In each case, all three bodies' orbits lay in the same plane. In some systems, all the bodies orbited in the same direction (direct); for other systems, one body orbited in the opposite direction from the other two (retrograde). The results of these experiments are presented in Chapter 4 (for retrograde systems) and Chapter 5 (for direct systems). The results are grouped in such a way that analytical curves may be fitted to the data. This allows predictions of stability lifetimes for similar systems without the need for lengthy numerical integration experiments. Systems whose masses, initial positions and initial velocities fall into certain ranges are always stable. These regions of hierarchical stability are mapped out and compared with corresponding regions of Hill stability. In the case of direct systems, commensurabilities give rise to large fluctuations in stability lifetimes, if the initial conditions are varied slightly. Additional statistical methods are described in Chapter 5 to cope with this effect. In Chapter 6, the results of Chapters 4 and 5 are compared with real three-body systems within the Solar System. Possible origins of the Solar System are discussed in the light of the results

Concepts for retractable roof structures.

Over the last decade there has been a worldwide increase in the use of retractable roofs for stadia. This increase has been based on the flexibility and better economic performance offered by venues featuring retractable roofs compared to those with traditional fixed roofs. With this increased interest an evolution in retractable roof systems has followed. This dissertation is concerned with the development of concepts for retractable roof systems. A review is carried out to establish the current state-of-the-art of retractable roof design. A second review of deployable structures is used to identify a suitable retractable structure for further development. The structure chosen is formed by a two-dimensional ring of pantographic bar elements interconnected through simple revolute hinges. A concept for retractable roofs is then proposed by covering the bar elements with rigid cover plates. To prevent the cover plates from inhibiting the motion of the structure a theorem governing the shape of these plate elements is developed through a geometrical study of the retractable mechanism. Applying the theorem it is found that retractable structures of any plan shape can be formed from plate elements only. To prove the concept a 1.3 meter diameter model is designed and built. To increase the structural efficiency of the proposed retractable roof concept it is investigated if the original plan shape can be adapted to a spherical surface. The investigation reveals that it is not possible to adapt the mechanism but the shape of the rigid cover plates can be adapted to a spherical surface. Three novel retractable mechanisms are then developed to allow opening and closing of a structure formed by such spherical plate elements. Two mechanisms are based on a spherical motion for the plate elements. It is shown that the spherical structure can be opened and closed by simply rotating the individual plates about fixed points. Hence a simple structure is proposed where each plate is rotated individually in a synchronous motion. To eliminate the need for mechanical synchronisation of the motion, a mechanism based on a reciprocal arrangement of the plates is developed. The plate elements are interconnected through sliding connections allowing them mutually to support each other, hence forming a self-supporting structure in which the motion of all plates is synchronised. To simplify the structure further, an investigation into whether the plate elements can be interconnected solely through simple revolute joints is carried out. This is not found to be possible for a spherical motion. However, a spatial mechanism is developed in which the plate elements are interconnected through bars and spherical joints. Geometrical optimisation of the motion path and connection points is used to eliminate the internal strains that occur in the initial design of this structure so a single degree-of-freedom mechanism is obtained. The research presented in this dissertation has hence led to the development of a series of novel concepts for retractable roof systems

Thermal Effects in Physics and Dynamics of Small Bodies of the Solar System

Thermal Effects in Physics and Dynamics of Small Bodies of the Solar System Abstract of the Ph.D. thesis \s David Capek It has been shown, that the thermal effects are very important in the dynamics of small Solar System bodies. A phenomenon which is known as the Yarkovsky effect is able to secularly change the semimajor axis of an orbit, while the YORP effect affects the rotation state of a body. The Yarkovsky effect and the YORP effect were previously calculated with many constraining assumptions like spherical shapes of asteroids, circular orbits, small variations of the surface temperature, principal axis rotation, constant thermal parameters, etc. We developed a sophisticated numerical model of the Yarkovsky/YORP effect without such simplifications. With this model, we have been able to describe the shape, the orbit, the rotation and the thermal behaviour of an asteroid in a very precise way. The YORP effect was studied on a sample of artificially generated shapes, roughly resembling Main Belt asteroids, and also on several shapes of real asteroids. The depen- dence of YORP on the obliquity and the thermal parameters of the surface were studied (Vokrouhlicky and Capek, 2002; Capek and Vokrouhlicky, 2004). A wide variety of pos- sible YORP evolution paths of the spin state was found. The possibility of...Tepelne jevy ve fyzice a dynamice malych teles slunecm soustavy Abstrakt dizertacnf prace David Capek Behem posledni doby se ukazalo, ze tepelne jevy jsou velmi vyznamne v dynamice malych teles slunecni soustavy. Intenzivne studovan byl predevslm jev zvany Jarkovskeho efekt, ktery je schopen dlouhodobe menit velkou poloosn drahy a YORP efekt, jez ovlivnuje rotacni stav telesa. Jarkovskeho a YORP efekt byly drive pocitany s mnoha omezujicimi predpoklady. Napnklad byly uvazovany kulove tvary asteroidu, kruhove drahy, male variace povrchove teploty, rotace okolo hlavni osy tenzoru setrvacnosti, konstantni tepelne parametry a podobne. Proto jsme vyvinnli numericky model pro vypocet Jarkovskeho/YORP jevu, ktery neni omezen temito predpoklady. S timto modelem jsme byli schopni velmi pfesne popsat tvar, drahu, rotaci a tepelne vlastnosti studovaneho telesa. YORP efekt byl studovan na vzorku umele vytvofenych tvaru} ktere odpovidaji aste- roidum hlavniho pasu, a take na tvarech skutecnych asteroidu. Zkoumali jsme zejmena savislost YORP jevu na obliquite a na tepelnych parametrech povrchu. Byla zjistena a diskutovana siroka skala moznosti v)'voje rotacniho stavu asteroidu (Vokrouhlickj'- and Capek, 2002; Capek and Vokrouhlickyj 2004). Pro nektere asteroidy bylo predpovezeno, ze Ize v budoucnosti ocekavat uspesnou...Institute of Theoretical PhysicsÚstav teoretické fyzikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Catalytic processes simulated at nano-scale: Growth of graphitic structures and functionalized graphene explained

Key dynamic processes at nano-scale, such as graphene hydrogenation and fluorination, and carbon nanotube (CNT) growth, cannot be observed in situ in real time. Nevertheless, such processes can be studied through complementary computational methods. This work simulates three important catalytic processes representing the growth of graphitic structures and functionalized graphene. The spillover phenomenon, which has been considered promising for efficient hydrogen storage, includes transfer of H from a metal catalyst to a graphitic receptor, to finally form a graphane island. Although, the spillover is energetically unfavorable to occur on pristine graphene, catalyst saturation provides a way for hydrogen adsorption on the receptor. Ab initio calculations show that the H chemical potential can be increased to a spillover favorable range. Unlike in graphane, upon graphene fluorination different stoichiometric phases form without a nucleation barrier, with the complete CF phase being thermodynamically most stable. After fluorination, graphene electronic properties are transformed from metallic to semiconducting. First-principles and tight-binding methods are used to investigate the patterning of nanoroads and quantum dots on these phases, combining metallic and semiconducting properties on the same sheet. In catalyzed CNT growth the metallic catalyst plays a fundamental role in cap nucleation. Such a mechanism cannot be seen in experiment, nor can it be simulated by first-principles due to its time-scale, yet it can be simulated through molecular dynamics. Tuning the metal-C interaction controls the condition for growth or encapsulation: Surface carbon-diffusion limits the growth below 600 K, and at higher temperatures they depend on cap lift-off. Such tuning can be done through catalyst alloying, as shown through ab initio simulations for Ni-Fe and Cu-Fe bimetallic catalysts. Catalyst shape also plays an important role in CNT growth. The minimization of the Ni surface energy defines the equilibrium crystal shape. Catalyst reshaping is analyzed through C adsorption by first-principles and reactive force fields. The Wulff-construction suggests a significant reduction of the surface energy anisotropy upon C adsorption, based on which a continuum phenomenological model that considers catalyst reshaping in CNT nucleation is formulated. This thesis explains the growth of graphitic structures and functionalized graphene at nano-scale through computational simulations

The contemporary visualization and modelling technologies and the techniques for the design of the green roofs

The contemporary design solutions are merging the boundaries between real and virtual world. The Landscape architecture like the other interdisciplinary field stepped in a contemporary technologies area focused on that, beside the good execution of works, designer solutions has to be more realistic and “touchable”. The opportunities provided by Virtual Reality are certainly not negligible, it is common knowledge that the designs in the world are already presented in this way so the Virtual Reality increasingly used. Following the example of the application of virtual reality in landscape architecture, this paper deals with proposals for the use of virtual reality in landscape architecture so that designers, clients and users would have a virtual sense of scope e.g. rooftop garden, urban areas, parks, roads, etc. It is a programming language that creates a series of images creating a whole, so certain parts can be controlled or even modified in VR. Virtual reality today requires a specific gadget, such as Occulus, HTC Vive, Samsung Gear VR and similar. The aim of this paper is to acquire new theoretical and practical knowledge in the interdisciplinary field of virtual reality, the ability to display using virtual reality methods, and to present through a brief overview the plant species used in the design and construction of an intensive roof garden in a Mediterranean climate, the basic characteristics of roofing gardens as well as the benefits they carry. Virtual and augmented reality as technology is a very powerful tool for landscape architects, when modeling roof gardens, parks, and urban areas. One of the most popular technologies used by landscape architects is Google Tilt Brush, which enables fast modeling. The Google Tilt Brush VR app allows modeling in three-dimensional virtual space using a palette to work with the use of a three dimensional brush. The terms of two "programmed" realities - virtual reality and augmented reality - are often confused. One thing they have in common, though, is VRML - Virtual Reality Modeling Language. In this paper are shown the ways on which this issue can be solved and by the way, get closer the term of Virtual Reality (VR), also all the opportunities which the Virtual reality offered us. As well, in this paper are shown the conditions of Mediterranean climate, the conceptual solution and the plant species which will be used by execution of intensive green roof on the motel “Marković”

The benefits of an additional practice in descriptive geomerty course: non obligatory workshop at the Faculty of Civil Engineering in Belgrade

At the Faculty of Civil Engineering in Belgrade, in the Descriptive geometry (DG) course, non-obligatory workshops named “facultative task” are held for the three generations of freshman students with the aim to give students the opportunity to get higher final grade on the exam. The content of this workshop was a creative task, performed by a group of three students, offering free choice of a topic, i.e. the geometric structure associated with some real or imagery architectural/art-work object. After the workshops a questionnaire (composed by the professors at the course) is given to the students, in order to get their response on teaching/learning materials for the DG course and the workshop. During the workshop students performed one of the common tests for testing spatial abilities, named “paper folding". Based on the results of the questionnairethe investigation of the linkages between:students’ final achievements and spatial abilities, as well as students’ expectations of their performance on the exam, and how the students’ capacity to correctly estimate their grades were associated with expected and final grades, is provided. The goal was to give an evidence that a creative work, performed by a small group of students and self-assessment of their performances are a good way of helping students to maintain motivation and to accomplish their achievement. The final conclusion is addressed to the benefits of additional workshops employment in the course, which confirmhigherfinal scores-grades, achievement of creative results (facultative tasks) and confirmation of DG knowledge adaption

Automated theory formation in pure mathematics

The automation of specific mathematical tasks such as theorem proving and algebraic manipulation have been much researched. However, there have only been a few isolated attempts to automate the whole theory formation process. Such a process involves forming new concepts, performing calculations, making conjectures, proving theorems and finding counterexamples. Previous programs which perform theory formation are limited in their functionality and their generality. We introduce the HR program which implements a new model for theory formation. This model involves a cycle of mathematical activity, whereby concepts are formed, conjectures about the concepts are made and attempts to settle the conjectures are undertaken.HR has seven general production rules for producing a new concept from old ones and employs a best first search by building new concepts from the most interesting old ones. To enable this, HR has various measures which estimate the interestingness of a concept. During concept formation, HR uses empirical evidence to suggest conjectures and employs the Otter theorem prover to attempt to prove a given conjecture. If this fails, HR will invoke the MACE model generator to attempt to disprove the conjecture by finding a counterexample. Information and new knowledge arising from the attempt to settle a conjecture is used to assess the concepts involved in the conjecture, which fuels the heuristic search and closes the cycle.The main aim of the project has been to develop our model of theory formation and to implement this in HR. To describe the project in the thesis, we first motivate the problem of automated theory formation and survey the literature in this area. We then discuss how HR invents concepts, makes and settles conjectures and how it assesses the concepts and conjectures to facilitate a heuristic search. We present results to evaluate HR in terms of the quality of the theories it produces and the effectiveness of its techniques. A secondary aim of the project has been to apply HR to mathematical discovery and we discuss how HR has successfully invented new concepts and conjectures in number theory

A Proposed Test of the Einstein Theory of Gravitation by Means of an Unshielded Orbiting Gyro Using Passive Telemetry

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB-07-67-C-0199National Aeronautics and Space Administration / NsG-44