Author: June Barrow-Green

Publisher: American Mathematical Soc.

ISBN: 9780821803677

Category: Mathematics

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## Poincaré and the Three Body Problem

The idea of chaos figures prominently in mathematics today. It arose in the work of one of the greatest mathematicians of the late 19th century, Henri Poincare, on a problem in celestial mechanics: the three body problem. This ancient problem - to describe the paths of three bodies in mutual gravitational interaction - is one of those which is simple to pose but impossible to solve precisely. Poincare's famous memoir on the three body problem arose from his entry in the competition celebrating the 60th birthday of King Oscar of Sweden and Norway. His essay won the prize and was set up in print as a paper in Acta Mathematica when it was found to contain a deep and critical error.In correcting this error Poincare discovered mathematical chaos, as is now clear from Barrow-Green's pioneering study of a copy of the original memoir annotated by Poincare himself, recently discovered in the Institut Mittag-Leffler in Stockholm. ""Poincare and the Three Body Problem"" opens with a discussion of the development of the three body problem itself and Poincare's related earlier work. The book also contains intriguing insights into the contemporary European mathematical community revealed by the workings of the competition. After an account of the discovery of the error and a detailed comparative study of both the original memoir and its rewritten version, the book concludes with an account of the final memoir's reception, influence and impact, and an examination of Poincare's subsequent highly influential work in celestial mechanics.
## The Three-Body Problem and the Equations of Dynamics

Here is an accurate and readable translation of a seminal article by Henri Poincaré that is a classic in the study of dynamical systems popularly called chaos theory. In an effort to understand the stability of orbits in the solar system, Poincaré applied a Hamiltonian formulation to the equations of planetary motion and studied these differential equations in the limited case of three bodies to arrive at properties of the equations’ solutions, such as orbital resonances and horseshoe orbits. Poincaré wrote for professional mathematicians and astronomers interested in celestial mechanics and differential equations. Contemporary historians of math or science and researchers in dynamical systems and planetary motion with an interest in the origin or history of their field will find his work fascinating.
## Linear Differential Equations and Group Theory from Riemann to Poincare

This book is a study of how a particular vision of the unity of mathematics, often called geometric function theory, was created in the 19th century. The central focus is on the convergence of three mathematical topics: the hypergeometric and related linear differential equations, group theory, and on-Euclidean geometry. The text for this second edition has been greatly expanded and revised, and the existing appendices enriched. The exercises have been retained, making it possible to use the book as a companion to mathematics courses at the graduate level.
## The Scientific Legacy of Poincaré

Henri Poincare (1854-1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. He created new mathematical branches, such as algebraic topology, dynamical systems, and automorphic functions, and he opened the way to complex analysis with several variables and to the modern approach to asymptotic expansions. He revolutionized celestial mechanics, discovering deterministic chaos. In physics, he is one of the fathers of special relativity, and his work in the philosophy of sciences is illuminating. For this book, about twenty world experts were asked to present one part of Poincare's extraordinary work. Each chapter treats one theme, presenting Poincare's approach, and achievements, along with examples of recent applications and some current prospects. Their contributions emphasize the power and modernity of the work of Poincare, an inexhaustible source of inspiration for researchers, as illustrated by the Fields Medal awarded in 2006 to Grigori Perelman for his proof of the Poincare conjecture stated a century before. This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, or physics, or more generally by anyone who likes mathematical and physical ideas. Rather than presenting detailed proofs, the main ideas are explained, and a bibliography is provided for those who wish to understand the technical details.
## Henri Poincaré

Henri Poincaré (1854-1912) was not just one of the most inventive, versatile, and productive mathematicians of all time--he was also a leading physicist who almost won a Nobel Prize for physics and a prominent philosopher of science whose fresh and surprising essays are still in print a century later. The first in-depth and comprehensive look at his many accomplishments, Henri Poincaré explores all the fields that Poincaré touched, the debates sparked by his original investigations, and how his discoveries still contribute to society today. Math historian Jeremy Gray shows that Poincaré's influence was wide-ranging and permanent. His novel interpretation of non-Euclidean geometry challenged contemporary ideas about space, stirred heated discussion, and led to flourishing research. His work in topology began the modern study of the subject, recently highlighted by the successful resolution of the famous Poincaré conjecture. And Poincaré's reformulation of celestial mechanics and discovery of chaotic motion started the modern theory of dynamical systems. In physics, his insights on the Lorentz group preceded Einstein's, and he was the first to indicate that space and time might be fundamentally atomic. Poincaré the public intellectual did not shy away from scientific controversy, and he defended mathematics against the attacks of logicians such as Bertrand Russell, opposed the views of Catholic apologists, and served as an expert witness in probability for the notorious Dreyfus case that polarized France. Richly informed by letters and documents, Henri Poincaré demonstrates how one man's work revolutionized math, science, and the greater world.
## Encyclopedia of Nonlinear Science

In 438 alphabetically-arranged essays, this work provides a useful overview of the core mathematical background for nonlinear science, as well as its applications to key problems in ecology and biological systems, chemical reaction-diffusion problems, geophysics, economics, electrical and mechanical oscillations in engineering systems, lasers and nonlinear optics, fluid mechanics and turbulence, and condensed matter physics, among others.
## Celestial Encounters

Celestial Encounters traces the history of attempts to solve the problem of celestial mechanics first posited in Isaac Newton's Principia in 1686. More generally, the authors reflect on mathematical creativity and the roles that chance encounters, politics, and circumstance play in it. 23 halftones. 64 line illustrations.
## Deleuze and the History of Mathematics

Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges are an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seeming incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it.
## Letzte Gedanken von Henri Poincaré

Der berühmte Mathematiker Henri Poincaré erwies sich als hervorragender Philosoph, dessen Werke tiefen Einfluss auf das menschschliche Denken ausübten. In diesem Buch stellt er unter anderen die Fragen nach der Veränderlichkeit der Naturgesetze, nach der Logik des Unendlichen, nach dem Grund für die Dreidimensionalität des Raums und nach der Sittlichkeit als Gemeingut.
## The Three-Body Problem

Cambridge, 1888. When schoolmistress Vanessa Duncan learns of a murder at St John's College, little does she know that she will become deeply entangled in the mystery. Dr Geoffrey Akers, Fellow in Pure Mathematics, has been found dead, struck down by a violent blow to the head. What could provoke such a brutal act? Vanessa, finding herself in amongst Cambridge's brightest scholarly minds, discovers that the motive may lie in mathematics itself. Drawn closer to the case by a blossoming friendship with mathematician Arthur Weatherburn, Vanessa begins to investigate. When she learns of Sir Isaac Newton's elusive 'n-body problem' and the prestigious prize offered to anyone with a solution, things begin to make sense. But with further deaths occurring and the threat of an innocent man being condemned, Vanessa must hurry with her calculations...
## Poincarés Vermutung

## Die drei Sonnen

Die Science-Fiction-Sensation aus China China, Ende der 1960er-Jahre: Während im ganzen Land die Kulturrevolution tobt, beginnt eine kleine Gruppe von Astrophysikern, Politkommissaren und Ingenieuren ein streng geheimes Forschungsprojekt. Ihre Aufgabe: Signale ins All zu senden und noch vor allen anderen Nationen Kontakt mit Außerirdischen aufzunehmen. Fünfzig Jahre später wird diese Vision Wirklichkeit – auf eine so erschreckende, umwälzende und globale Weise, dass dieser Kontakt das Schicksal der Menschheit für immer verändern wird.
## Three Body Dynamics and Its Applications to Exoplanets

This brief book provides an overview of the gravitational orbital evolution of few-body systems, in particular those consisting of three bodies. The authors present the historical context that begins with the origin of the problem as defined by Newton, which was followed up by Euler, Lagrange, Laplace, and many others. Additionally, they consider the modern works from the 20th and 21st centuries that describe the development of powerful analytical methods by Poincare and others. The development of numerical tools, including modern symplectic methods, are presented as they pertain to the identification of short-term chaos and long term integrations of the orbits of many astronomical architectures such as stellar triples, planets in binaries, and single stars that host multiple exoplanets. The book includes some of the latest discoveries from the Kepler and now K2 missions, as well as applications to exoplanets discovered via the radial velocity method. Specifically, the authors give a unique perspective in relation to the discovery of planets in binary star systems and the current search for extrasolar moons.
## Der Wert der Wissenschaft

## A History in Sum

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard's mathematics department was at the center of these developments. A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics--in algebraic geometry, complex analysis, and other esoteric subdisciplines that are rarely written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose. The story begins in 1825, when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics--an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce's successors transformed the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff. Influential figures soon flocked to Harvard, some overcoming great challenges to pursue their elected calling. A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond.
## Mathematics Unbound

Although today's mathematical research community takes its international character very much for granted, this ''global nature'' is relatively recent, having evolved over a period of roughly 150 years-from the beginning of the nineteenth century to the middle of the twentieth century. During this time, the practice of mathematics changed from being centered on a collection of disparate national communities to being characterized by an international group of scholars for whom thegoal of mathematical research and cooperation transcended national boundaries. Yet, the development of an international community was far from smooth and involved obstacles such as war, political upheaval, and national rivalries. Until now, this evolution has been largely overlooked by historians andmathematicians alike. This book addresses the issue by bringing together essays by twenty experts in the history of mathematics who have investigated the genesis of today's international mathematical community. This includes not only developments within component national mathematical communities, such as the growth of societies and journals, but also more wide-ranging political, philosophical, linguistic, and pedagogical issues. The resulting volume is essential reading for anyone interestedin the history of modern mathematics. It will be of interest to mathematicians, historians of mathematics, and historians of science in general.
## Proceedings of the Canadian Society for the History and Philosophy of Mathematics

## History of Topology

Topology, for many years, has been one of the most exciting and influential fields of research in modern mathematics. Although its origins may be traced back several hundred years, it was Poincaré who "gave topology wings" in a classic series of articles published around the turn of the century. While the earlier history, sometimes called the prehistory, is also considered, this volume is mainly concerned with the more recent history of topology, from Poincaré onwards. As will be seen from the list of contents the articles cover a wide range of topics. Some are more technical than others, but the reader without a great deal of technical knowledge should still find most of the articles accessible. Some are written by professional historians of mathematics, others by historically-minded mathematicians, who tend to have a different viewpoint.
## Zdenek Kopal's Binary Star Legacy

An international conference entitled "Zdenek Kopal's Binary Star Legacy" was held on the occasion of the late Professor Kopal's 90th birthday in his home town of Litomyšl/Czech Republic and dedicated to the memory of one of the leading astronomers of the 20th century. Professor Kopal, who devoted 60 years of his scientific life to the exploration of close binary systems, initiated a breakthrough in this field with his description of binary components as non-spherical stars deformed by gravity, with surfaces following Roche equipotentials. Such knowledge triggered the development of new branches of astrophysics dealing with the structure and evolution of close binaries and the interaction effects displayed by exciting objects such as cataclysmic variables, symbiotic stars or X-ray binaries. Contributions to this conference included praise of the achievements of a great astronomer and personal reminiscences brought forward by Kopal's former students and colleagues, and reflected the state of the art of the dynamically evolving field of binary research, which owes so much to the pioneering work of Zdenek Kopal.
## Wissenschaft und Methode

I. Forscher und Wissenschaftler: Die Auswahl der Tatsachen / Die Zukunft der Mathematik / Die mathematische Erfindung / Der Zufall II. Die mathematische Schlußweise: Die Relativität des Raumes / Die mathematischen Definitionen und der Unterricht / Mathematik und Logik / Die neue Logik / Die neuesten Arbeiten der Logistiker III. Die neue Mechanik: Mechanik und Radium / Mechanik und Optik / Die neue Mechanik und die Astronomie IV. Die Wissenschaft der Astronomie: Milchstraße und Gastheorie / Die Geodäsie in Frankreich Erläuternde Anmerkungen (von F. Lindemann) "Viele Mathematiker glauben, daß man die Mathematik auf die Gesetze der formalen Logik zurückführen kann. Unerhörte Anstrengungen wurden zu diesem Zwecke unternommen; zur Erreichung des bezeichneten Zieles scheute man sich z.B. nicht, die historische Ordnung in der Entstehung unserer Vorstellungen umzukehren, und man suchte das Endliche durch das Unendliche zu erklären. Für alle, welche das Problem ohne Voreingenommenheit angereifen, glaube ich im folgenden gezeigt zu haben, daß diesem Bestreben eine trügerische Illusion zugrunde liegt. Wie ich hoffe, wird der Leser die Wichtigkeit der Frage verstehen [...]." Henri Poincaré

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Author: June Barrow-Green

Publisher: American Mathematical Soc.

ISBN: 9780821803677

Category: Mathematics

Page: 272

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