Thesis. 1979. M.S.--Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERONAUTICS. / Includes bibliographical references. / M.S.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERO. / Vita. / Includes bibliographical references. / by Andrew Joseph Green. / Ph.D.

We investigate the existence of a symmetric singular periodic brake orbit in the equal mass, fully symmetric planar four body problem. Using regularized coordinates, we remove the singularity of binary collision for each symmetric pair. We use topological and symmetry tools in our investigation.

In this dissertation, we devise a quantization scheme to construct eigenfunctions by classical periodic orbits in both regular systems as well as chaotic systems. Our method is based on the principle that eigenfunctions can be resolved from a time-dependent wavefunction. This is different from the classical (or EBK) quantization scheme that constructs eigenfunction in the energy-domain. The advantage of our method is that it can be applied to more varieties of systems, including some chaotic systems. Three systems, the simple harmonic oscillator, the x⁴-potential oscillator, and the x²y² quartic-oscillator, are used as examples for our eigenfunction construction. The key to the constructions is a family (or families) of periodic orbits with a newly defined quantization rule, the resolving quantization rule. The eigenspectrum for the x⁴-potential oscillator is also computed. Furthermore, the classical Green's function is used to explain the relation between the resolving quantization rule and the classical quantization rule. This dissertation begins with an introduction in Chapter 1. The semiclassical theory for the eigenfunction construction by periodic orbits is developed in Chapter 2. In Chapter 3 and Chapter 4, eigenfunctions are constructed for the simple harmonic oscillator, the x⁴-potential oscillator, and the x²y² quartic-oscillator. The eigenspectrum for the x⁴-potential oscillator is computed in Chapter 5. Chapter 6 is devoted to discussions including the interpretation of the resolving quantization rule from the classical Green's function, the interpretation of the photoabsorption spectrum for a Rydberg atom in a magnetic field, and the comparison of our method with the EBK quantization scheme. Conclusions are made in Chapter 7. / text

published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

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In this thesis, we classify finite orbits of the action of the pure braid group over a certain large open subset of the SL(2,C) character variety of the Riemann sphere with five boundary components, i.e. Σ5. This problem arises in the context of classifying algebraic solutions of the Garnier system G2, that is the two variable analogue of the famous sixth Painleve equation PVI. The structure of the analytic continuation of these solutions is described in terms of the action of the pure braid group on the fundamental group of Σ5. To deal with this problem, we introduce a system of co-adjoint coordinates on a big open subset of the SL(2,C) character variety of Σ5. Our classifica- tion method is based on the definition of four restrictions of the action of the pure braid group such that they act on some of the co-adjoint coordi- nates of Σ5 as the pure braid group acts on the co-adjoint coordinates of the character variety of the Riemann sphere with four boundary components, i.e. Σ4, for which the classification of all finite orbits is known. In order to avoid redundant elements in our final list, a group of symmetries G of the large open subset is introduced and the final classification is achieved modulo the action of G. We present a final list of 54 finite orbits.

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-10Bitstream added on 2014-06-13T18:53:34Z : No. of bitstreams: 1 santos_pmp_me_guara.pdf: 4359750 bytes, checksum: 85e5035af4f1d67cd7f1b2afa6bc956a (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho analisamos numericamente a região externa do sistema Plutão-Caronte através da insercão de partículas-teste inicialmente em ´orbitas do tipo-P prógradas e retrógradas, no sistema formado por Plutão, Caronte, Nix e Hidra. Destas integracões numéricas foram geradas grades semi-eixo maior em função da excentricidade definindo-se regiões de partículas em orbitas estáveis e regiões de colisão e escape. Na vizinhança dos satélites Nix e Hidra foram identificadas regiões caóticas, em que partículas localizadas dentro desta região têm suas excentricidades e semi-eixo maiores aumentados e escapam ou colidem com um corpo massivo do sistema. Um conjunto de partículas permaneceram em regiões próximas das orbitas de Nix e Hidra, possivelmente coorbitais de Nix e Hidra. Para ambos os casos, prógrado e retrógrado, a região estável”é maior na região externa do sistema, após a órbita de Hidra, dependendo do valor da excentricidade. Também foram realizadas simulações numéricas inserindo satélites hipotéticos massivos além da órbita de Caronte e os efeitos causados nas órbitas de Nix e Hidra foram analisados. Um estudo numérico preliminar dos efeitos da Press˜ao de Radiac¸ ˜ao Solar em partículas com raios de 1μm, 3μm, 5μm e 10μm foi realizado. Este estudo mostrou que partículas sob os efeitos do Arrasto de Poynting-Robertson deca´ıram em 1,45 × 106 anos (partículas de 1μm de raio) e 1,45 × 107 anos (partíıculas de 10μm de raio), enquanto que a Pressão de Radiac¸ ˜ao causou variacões das excentricidades das partículas fazendo com que em alguns casos houvesse colisões com o planeta. / In this work we performed a numerical analysis of the the outer region of the Pluto-Charon system by the insertion of a sample of test particles initially in P-type prograde and retrograde orbits, in the system formed by Pluto, Charon, Nix and Hydra. These numerical integrations generated diagrams of semi-major axis versus eccentricity which define regions of particles in stable orbits and regions of collision and escape. In the vicinity of the satellites Nix and Hydra were identified chaotic regions, where particles located in this region have their eccentricities and semi-major axis increased provoking an ejection or collision with a massive body of the system. A set of particles remained in regions near the orbits of Nix and Hydra, possibly coorbitais with them. For both cases, prograde and retrograde, the “stable” region is larger in the outer region of the system, after Hydra’s orbit, depending on the value of eccentricity. Numerical simulations were also performed by inserting some massive hypothetical satellites beyond the Charon’s orbit and the effects on the orbits of Nix and Hydra were analyzed. A preliminary numerical study of the effects of the solar radiation force on a sample of particles with radii of 1μm, 3μm, 5μm e 10μm was performed. This study showed that particles under the effects of the Poynting-Robertson drag decay on a time scale between 1.45×106 years (particles of 1μm in radius) and 1.45×107 years (particles of 10μm in radius), while the radiation pressure caused variations of the eccentricities of the particles causing in some cases collisions with the planet.