Essa dissertação trata de geometria simplética e suas aplicações, apresentando conceitos tais como o gradiente simplético e também o teorema de Darboux. Discutimos a formulação Lagrangeana da mecânica, apresentando as equações de Euler-Lagrange e, usando a geometria simplética, mostramos como estes naturalmente evoluem para o formalismo Hamiltoneano e as equações de Hamilton. Introduzimos também o conceito da métrica de Jacobi e demonstramos o teorema de Noether. Apresentamos o conceito de ações simpléticas e Hamiltoneanas, bem como aplicações momento e comomento. São demonstrados resultados importantes como o teorema de Kirillov-Kostant-Sourieau para órbitas coadjuntas e a redução simplética de Marsden-Weinstein-Meyer. Os resultados centrais apresentados são o teorema de Atiyah-Guillemin-Steinberg de convexidade, o teorema de Schur e Horn para matrizes unitárias e o teorema de Delzant, este último sendo apresentado apenas com uma ideia da prova. / This thesis is about symplectic geometry and its applications, presenting concepts such as the symplectic gradient and also Darboux\'s theorem. We discuss the Lagrangian formulation of mechanics, presenting the Euler-Lagrange equations and, using symplectic geometry, show how those naturally evolve into the Hamiltonian formalism and the Hamilton equations. We instroduce also the concept of the Jacobi metrics and prove Noether\'s theorem. We also introduce the concept of symplectic and Hamiltonian actions as well as moment and comoment maps. We prove important results such as the Kirillov-Kostant-Sourieau theorem for coadjoint orbits and the symplectic reduction of Marsden-Weinstein-Meyer. The central results presented are the convexity theorem of Guillemin-Atiyah-Steinberg, the Schur and Horn theorem for unitary matrices and the Delzant theorem, this last one being presented only with an idea of the proof.

In the early 1980s a landmark result was obtained by Atiyah and independently Guillemin and Sternberg: the image of the momentum map for a torus action on a compact symplectic manifold is a convex polyhedron. Atiyah's proof makes use of the fact that level sets of the momentum map are connected. These proofs work in the setting of finite-dimensional compact symplectic manifolds. One can ask how these results generalize. A well-known example of an infinite-dimensional symplectic manifold with a finite-dimensional torus action is the based loop group. Atiyah and Pressley proved convexity for this example, but not connectedness of level sets. A proof of connectedness of level sets for the based loop group was provided by Harada, Holm, Jeffrey and Mare in 2006. In this thesis we study Hilbert manifolds equipped with a strong symplectic structure and a finite-dimensional group action preserving the strong symplectic structure. We prove connectedness of regular generic level sets of the momentum map. We use this to prove convexity of the image of the momentum map.

In the early 1980s a landmark result was obtained by Atiyah and independently Guillemin and Sternberg: the image of the momentum map for a torus action on a compact symplectic manifold is a convex polyhedron. Atiyah's proof makes use of the fact that level sets of the momentum map are connected. These proofs work in the setting of finite-dimensional compact symplectic manifolds. One can ask how these results generalize. A well-known example of an infinite-dimensional symplectic manifold with a finite-dimensional torus action is the based loop group. Atiyah and Pressley proved convexity for this example, but not connectedness of level sets. A proof of connectedness of level sets for the based loop group was provided by Harada, Holm, Jeffrey and Mare in 2006. In this thesis we study Hilbert manifolds equipped with a strong symplectic structure and a finite-dimensional group action preserving the strong symplectic structure. We prove connectedness of regular generic level sets of the momentum map. We use this to prove convexity of the image of the momentum map.

Essa dissertação trata de geometria simplética e suas aplicações, apresentando conceitos tais como o gradiente simplético e também o teorema de Darboux. Discutimos a formulação Lagrangeana da mecânica, apresentando as equações de Euler-Lagrange e, usando a geometria simplética, mostramos como estes naturalmente evoluem para o formalismo Hamiltoneano e as equações de Hamilton. Introduzimos também o conceito da métrica de Jacobi e demonstramos o teorema de Noether. Apresentamos o conceito de ações simpléticas e Hamiltoneanas, bem como aplicações momento e comomento. São demonstrados resultados importantes como o teorema de Kirillov-Kostant-Sourieau para órbitas coadjuntas e a redução simplética de Marsden-Weinstein-Meyer. Os resultados centrais apresentados são o teorema de Atiyah-Guillemin-Steinberg de convexidade, o teorema de Schur e Horn para matrizes unitárias e o teorema de Delzant, este último sendo apresentado apenas com uma ideia da prova. / This thesis is about symplectic geometry and its applications, presenting concepts such as the symplectic gradient and also Darboux\'s theorem. We discuss the Lagrangian formulation of mechanics, presenting the Euler-Lagrange equations and, using symplectic geometry, show how those naturally evolve into the Hamiltonian formalism and the Hamilton equations. We instroduce also the concept of the Jacobi metrics and prove Noether\'s theorem. We also introduce the concept of symplectic and Hamiltonian actions as well as moment and comoment maps. We prove important results such as the Kirillov-Kostant-Sourieau theorem for coadjoint orbits and the symplectic reduction of Marsden-Weinstein-Meyer. The central results presented are the convexity theorem of Guillemin-Atiyah-Steinberg, the Schur and Horn theorem for unitary matrices and the Delzant theorem, this last one being presented only with an idea of the proof.

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Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-29T14:01:29Z No. of bitstreams: 1 genaropablozamudiochauca.pdf: 295489 bytes, checksum: 18212d3cbd798de7a3d5a0a546393c3c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-29T19:44:50Z (GMT) No. of bitstreams: 1 genaropablozamudiochauca.pdf: 295489 bytes, checksum: 18212d3cbd798de7a3d5a0a546393c3c (MD5) / Made available in DSpace on 2017-05-29T19:44:50Z (GMT). No. of bitstreams: 1 genaropablozamudiochauca.pdf: 295489 bytes, checksum: 18212d3cbd798de7a3d5a0a546393c3c (MD5) Previous issue date: 2012-03-29 / CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico / Em este trabalho estudamos algumas estruturas matemáticas presentes no modelo semiclássico para o spin não relativístico proposto nas referências [5] e [6]. Obtemos as equações semiclássicas de movimento para o spin não relativístico aplicando o teorema de Ehrenfest à equação de Pauli. Olhando o spin S como um momento angular interno, identicamos ele como a aplicação de momento ligada à ação de Poisson de SO(3) sobre o espaço de fase interno R6. Para eliminar os graus de liberdade extras presentes no modelo restringimos a dinâmica a uma superfície de spin V3 impondo vínculos. Além disso, mostramos que a superfície de spin V3 tem estrutura de fibrado com base S2, fibra típica SO(2) e com aplicação de projeção S. Finalmente apresentamos a formulação do problema variacional para o modelo. / In this work we study some mathematical structures arising in a nonrelativistic spinningparticle model proposed in [5] and [6]. We obtain the semiclassical equations of motion from the Pauli equation via the Ehrenfest theorem. Looking for the spin S as an intrisic angular momentum, we identify it with the momentum map of the SO(3) Poisson action on the inner phase space R6. In order to eliminate the extra degrees of freedom, we impose some constraints which restrict the evolution of the system on the spin surface V3. We show that V3 is a fiber bundle with base S2, standard fiber SO(2) and projection S. Finally, we present the formulation of variational problem for the model.