New Mathematical Methods for Physics (Computational Mathematics and Analysis)

Pommaret, Jean-francois

ISBN 10: 1536134104 / ISBN 13: 9781536134100

Published by Nova Science Pub Inc, 2018

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Bibliographic Details

• Title: New Mathematical Methods for Physics (Computational Mathematics and Analysis)
• Publisher: Nova Science Pub Inc
• Publication Date: 2018
• Language: English
• ISBN 10: 1536134104
• ISBN 13: 9781536134100
• Binding: paperback
• Condition: New
• Book Type: book

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Synopsis

The concept of “group” has been introduced in mathematics for the first time by E. Galois (1830) and slowly passed from algebra to geometry with the work of S. Lie on Lie groups (1880) and Lie pseudogroups (1890) of transformations. The concept of a finite length differential sequence, now called the Janet sequence, had been described for the first time by M. Janet (1920). Then, the work of D. C. Spencer (1970) has been the first attempt to use the formal theory of systems of partial differential equations (PDE) in order to study the formal theory of Lie pseudogroups. However, the linear and nonlinear Spencer sequences for Lie pseudogroups, though never used in physics, largely supersede the ”Cartan structure equations ” (1905) and are quite different from the ”Vessiot structure equations ” (1903), introduced for the same purpose but never acknowledged by E. Cartan or successors. Meanwhile, mixing differential geometry with homological algebra, M. Kashiwara (1970) created “algebraic analysis” in order to study differential modules and double duality. By chance, unexpected arguments have been introduced by the brothers E. and F. Cosserat (1909) in order to revisit elasticity and by H. Weyl (1918) in order to revisit electromagnetism through a unique differential sequence only depending on the structure of the conformal group of space-time. The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of the extension. The purpose of this book is first to revisit Gauge Theory and General Relativity in light of the latest developments just described and then to apply the differential Galois theory in order to revisit various domains of mechanics (Shell theory, Chain theory, Frenet-Serret formulas, Hamilton-Jacobi equations). All the results presented are new. (Nova)