226 research outputs found
The degenerate C. Neumann system I: symmetry reduction and convexity
The C. Neumann system describes a particle on the sphere S^n under the
influence of a potential that is a quadratic form. We study the case that the
quadratic form has l+1 distinct eigenvalues with multiplicity. Each group of
m_\sigma equal eigenvalues gives rise to an O(m_\sigma)-symmetry in
configuration space. The combined symmetry group G is a direct product of l+1
such factors, and its cotangent lift has an Ad^*-equivariant Momentum mapping.
Regular reduction leads to the Rosochatius system on S^l, which has the same
form as the Neumann system albeit for an additional effective potential.
To understand how the reduced systems fit together we use singular reduction
to construct an embedding of the reduced Poisson space T^*{S^n}/G into
R^{3l+3}$. The global geometry is described, in particular the bundle structure
that appears as a result of the superintegrability of the system. We show how
the reduced Neumann system separates in elliptical-spherical co-ordinates. We
derive the action variables and frequencies as complete hyperelliptic integrals
of genus l. Finally we prove a convexity result for the image of the Casimir
mapping restricted to the energy surface.Comment: 36 page
Luca Pacioli and his 1500 book de Viribus Quantitatis
Tese de mestrado, História e Filosofia das Ciências, Universidade de Lisboa, Faculdade de Ciências, 2015As the field grows, History of Science has become wider-ranging than a purely progress-oriented view of the history of Science. The History of Mathematics, even though more resilient, has shown to follow the same development. The present dissertation tries to contribute to the general study by shedding some light on a book which has been belittled, misinterpreted or ignored altogether, De Viribus Quantitatis, one of the major historical recreational mathematics books, and its author Luca Pacioli. This text aims to provide a modern updated survey of the content of this book for related studies, as well as a résumé of its contents.Com o crescimento do ramo de História das Ciências este tem vindo a desenvolver um olhar mais abrangente que a clássica visão dedicada ao progresso das ideias cientÃficas. A História da Matemática, embora mais resiliente, também tem vindo a mostrar interesse em expandir os seus horizontes. No presente texto tentamos contribuir para o estudo geral destas disciplinas estudando um tratado que pouca atenção tem tido até ao momento, sendo até mesmo mal interpretado. Trata-se De Viribus Quantitatis, sendo este um dos maiores compêndio de matemática recreativa no seu contexto histórico. O seu autor, Luca Pacioli, sendo uma personalidade de grande interesse e mais conhecido por outras obras suas. Nestas páginas tentamos fornecer uma versão atualisada da documentação relativa ao tratado tal como um resumo dos seus conteúdos
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