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Author: Dwight E. Neuenschwander Publisher: JHU Press ISBN: 1421422689 Category : Science Languages : en Pages : 338
Book Description
One of the most important—and beautiful—mathematical solutions ever devised, Noether’s theorem touches on every aspect of physics. "In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began."—Albert Einstein The year was 1915, and the young mathematician Emmy Noether had just settled into Göttingen University when Albert Einstein visited to lecture on his nearly finished general theory of relativity. Two leading mathematicians of the day, David Hilbert and Felix Klein, dug into the new theory with gusto, but had difficulty reconciling it with what was known about the conservation of energy. Knowing of her expertise in invariance theory, they requested Noether’s help. To solve the problem, she developed a novel theorem, applicable across all of physics, which relates conservation laws to continuous symmetries—one of the most important pieces of mathematical reasoning ever developed. Noether’s “first” and “second” theorem was published in 1918. The first theorem relates symmetries under global spacetime transformations to the conservation of energy and momentum, and symmetry under global gauge transformations to charge conservation. In continuum mechanics and field theories, these conservation laws are expressed as equations of continuity. The second theorem, an extension of the first, allows transformations with local gauge invariance, and the equations of continuity acquire the covariant derivative characteristic of coupled matter-field systems. General relativity, it turns out, exhibits local gauge invariance. Noether’s theorem also laid the foundation for later generations to apply local gauge invariance to theories of elementary particle interactions. In Dwight E. Neuenschwander’s new edition of Emmy Noether’s Wonderful Theorem, readers will encounter an updated explanation of Noether’s “first” theorem. The discussion of local gauge invariance has been expanded into a detailed presentation of the motivation, proof, and applications of the “second” theorem, including Noether’s resolution of concerns about general relativity. Other refinements in the new edition include an enlarged biography of Emmy Noether’s life and work, parallels drawn between the present approach and Noether’s original 1918 paper, and a summary of the logic behind Noether’s theorem.
Author: Dwight E. Neuenschwander Publisher: JHU Press ISBN: 1421422689 Category : Science Languages : en Pages : 338
Book Description
One of the most important—and beautiful—mathematical solutions ever devised, Noether’s theorem touches on every aspect of physics. "In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began."—Albert Einstein The year was 1915, and the young mathematician Emmy Noether had just settled into Göttingen University when Albert Einstein visited to lecture on his nearly finished general theory of relativity. Two leading mathematicians of the day, David Hilbert and Felix Klein, dug into the new theory with gusto, but had difficulty reconciling it with what was known about the conservation of energy. Knowing of her expertise in invariance theory, they requested Noether’s help. To solve the problem, she developed a novel theorem, applicable across all of physics, which relates conservation laws to continuous symmetries—one of the most important pieces of mathematical reasoning ever developed. Noether’s “first” and “second” theorem was published in 1918. The first theorem relates symmetries under global spacetime transformations to the conservation of energy and momentum, and symmetry under global gauge transformations to charge conservation. In continuum mechanics and field theories, these conservation laws are expressed as equations of continuity. The second theorem, an extension of the first, allows transformations with local gauge invariance, and the equations of continuity acquire the covariant derivative characteristic of coupled matter-field systems. General relativity, it turns out, exhibits local gauge invariance. Noether’s theorem also laid the foundation for later generations to apply local gauge invariance to theories of elementary particle interactions. In Dwight E. Neuenschwander’s new edition of Emmy Noether’s Wonderful Theorem, readers will encounter an updated explanation of Noether’s “first” theorem. The discussion of local gauge invariance has been expanded into a detailed presentation of the motivation, proof, and applications of the “second” theorem, including Noether’s resolution of concerns about general relativity. Other refinements in the new edition include an enlarged biography of Emmy Noether’s life and work, parallels drawn between the present approach and Noether’s original 1918 paper, and a summary of the logic behind Noether’s theorem.
Author: Yvette Kosmann-Schwarzbach Publisher: Springer Science & Business Media ISBN: 0387878688 Category : Mathematics Languages : en Pages : 205
Book Description
In 1915 and 1916 Emmy Noether was asked by Felix Klein and David Hilbert to assist them in understanding issues involved in any attempt to formulate a general theory of relativity, in particular the new ideas of Einstein. She was consulted particularly over the difficult issue of the form a law of conservation of energy could take in the new theory, and she succeeded brilliantly, finding two deep theorems. But between 1916 and 1950, the theorem was poorly understood and Noether's name disappeared almost entirely. People like Klein and Einstein did little more then mention her name in the various popular or historical accounts they wrote. Worse, earlier attempts which had been eclipsed by Noether's achievements were remembered, and sometimes figure in quick historical accounts of the time. This book carries a translation of Noether's original paper into English, and then describes the strange history of its reception and the responses to her work. Ultimately the theorems became decisive in a shift from basing fundamental physics on conservations laws to basing it on symmetries, or at the very least, in thoroughly explaining the connection between these two families of ideas. The real significance of this book is that it shows very clearly how long it took before mathematicians and physicists began to recognize the seminal importance of Noether's results. This book is thoroughly researched and provides careful documentation of the textbook literature. Kosmann-Schwarzbach has thus thrown considerable light on this slow dance in which the mathematical tools necessary to study symmetry properties and conservation laws were apparently provided long before the orchestra arrives and the party begins.
Author: James Read Publisher: Cambridge University Press ISBN: 1108786812 Category : Science Languages : en Pages : 388
Book Description
In 1918, Emmy Noether, in her paper Invariante Variationsprobleme, proved two theorems (and their converses) on variational problems that went on to revolutionise theoretical physics. 100 years later, the mathematics of Noether's theorems continues to be generalised, and the physical applications of her results continue to diversify. This centenary volume brings together world-leading historians, philosophers, physicists, and mathematicians in order to clarify the historical context of this work, its foundational and philosophical consequences, and its myriad physical applications. Suitable for advanced undergraduate and graduate students and professional researchers, this is a go-to resource for those wishing to understand Noether's work on variational problems and the profound applications which it finds in contemporary physics.
Author: Gennadi Sardanashvily Publisher: Springer ISBN: 9462391718 Category : Mathematics Languages : en Pages : 297
Book Description
The book provides a detailed exposition of the calculus of variations on fibre bundles and graded manifolds. It presents applications in such area's as non-relativistic mechanics, gauge theory, gravitation theory and topological field theory with emphasis on energy and energy-momentum conservation laws. Within this general context the first and second Noether theorems are treated in the very general setting of reducible degenerate graded Lagrangian theory.
Author: P.G.L. Leach Publisher: MDPI ISBN: 3039282344 Category : Science Languages : en Pages : 186
Book Description
In Noether's original presentation of her celebrated theorem of 1918, allowances were made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to only point transformations. In recent decades, this diminution of the power of Noether's Theorem has been partly countered, in particular, in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look at the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables.
Author: Dwight E. Neuenschwander Publisher: JHU Press ISBN: 1421422670 Category : Mathematics Languages : en Pages : 338
Book Description
"Dwight E. Neuenschwander's introduction to the theorem's genesis, applications, and consequences artfully unpacks its universal importance and unsurpassed elegance. Drawing from over thirty years of teaching the subject, Neuenschwander uses mechanics, optics, geometry, and field theory to point the way to a deep understanding of Noether's Theorem. The three sections provide a step-by-step, simple approach to the less-complex concepts surrounding the theorem, in turn instilling the knowledge and confidence needed to grasp the full wonder it encompasses. Illustrations and worked examples throughout each chapter serve as signposts on the way to this apex of physics."--Publisher's description.
Author: Gary Webb Publisher: Springer ISBN: 3319725114 Category : Science Languages : en Pages : 301
Book Description
This text focuses on conservation laws in magnetohydrodynamics, gasdynamics and hydrodynamics. A grasp of new conservation laws is essential in fusion and space plasmas, as well as in geophysical fluid dynamics; they can be used to test numerical codes, or to reveal new aspects of the underlying physics, e.g., by identifying the time history of the fluid elements as an important key to understanding fluid vorticity or in investigating the stability of steady flows. The ten Galilean Lie point symmetries of the fundamental action discussed in this book give rise to the conservation of energy, momentum, angular momentum and center of mass conservation laws via Noether’s first theorem. The advected invariants are related to fluid relabeling symmetries – so-called diffeomorphisms associated with the Lagrangian map – and are obtained by applying the Euler-Poincare approach to Noether’s second theorem. The book discusses several variants of helicity including kinetic helicity, cross helicity, magnetic helicity, Ertels’ theorem and potential vorticity, the Hollman invariant, and the Godbillon Vey invariant. The book develops the non-canonical Hamiltonian approach to MHD using the non-canonical Poisson bracket, while also refining the multisymplectic approach to ideal MHD and obtaining novel nonlocal conservation laws. It also briefly discusses Anco and Bluman’s direct method for deriving conservation laws. A range of examples is used to illustrate topological invariants in MHD and fluid dynamics, including the Hopf invariant, the Calugareanu invariant, the Taylor magnetic helicity reconnection hypothesis for magnetic fields in highly conducting plasmas, and the magnetic helicity of Alfvén simple waves, MHD topological solitons, and the Parker Archimedean spiral magnetic field. The Lagrangian map is used to obtain a class of solutions for incompressible MHD. The Aharonov-Bohm interpretation of magnetic helicity and cross helicity is discussed. In closing, examples of magnetosonic N-waves are used to illustrate the role of the wave number and group velocity concepts for MHD waves. This self-contained and pedagogical guide to the fundamentals will benefit postgraduate-level newcomers and seasoned researchers alike.
Author: A.J. Kox Publisher: Springer Science & Business Media ISBN: 0817644547 Category : Science Languages : en Pages : 384
Book Description
Outgrowth of 6th Int'l Conference on the History of General Relativity, held in Amsterdam on June 26-29, 2002 Contributions from notable experts offer both new and historical insights on gravitation, general relativity, cosmology, unified field theory, and the history of science Topics run gamet from detailed mathematical discussions to more personal recollections of relativity as seen through the eyes of the public and renowned relativists
Author: Arkady Plotnitsky Publisher: Springer Nature ISBN: 3031136780 Category : Mathematics Languages : en Pages : 307
Book Description
This book is a philosophical study of mathematics, pursued by considering and relating two aspects of mathematical thinking and practice, especially in modern mathematics, which, having emerged around 1800, consolidated around 1900 and extends to our own time, while also tracing both aspects to earlier periods, beginning with the ancient Greek mathematics. The first aspect is conceptual, which characterizes mathematics as the invention of and working with concepts, rather than only by its logical nature. The second, Pythagorean, aspect is grounded, first, in the interplay of geometry and algebra in modern mathematics, and secondly, in the epistemologically most radical form of modern mathematics, designated in this study as radical Pythagorean mathematics. This form of mathematics is defined by the role of that which beyond the limits of thought in mathematical thinking, or in ancient Greek terms, used in the book’s title, an alogon in the logos of mathematics. The outcome of this investigation is a new philosophical and historical understanding of the nature of modern mathematics and mathematics in general. The book is addressed to mathematicians, mathematical physicists, and philosophers and historians of mathematics, and graduate students in these fields.
Author: Dwight E. Neuenschwander
Author: Dwight E. Neuenschwander
Author: Yvette Kosmann-Schwarzbach
Author: James Read
Author: Gennadi Sardanashvily
Author: James Read
Author: P.G.L. Leach
Author: Dwight E. Neuenschwander
Author: Gary Webb
Author: A.J. Kox
Author: Arkady Plotnitsky