Poincaré and the Three Body Problem

Author: June Barrow-Green

Publisher: American Mathematical Soc.

ISBN: 9780821803677

Category: Mathematics

Page: 272

View: 1786


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

Poincaré’s Foundational Work on Dynamical Systems Theory

Author: Henri Poincaré

Publisher: Springer

ISBN: 3319528998

Category: Mathematics

Page: 248

View: 319


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

Author: Jeremy Gray

Publisher: Springer Science & Business Media

ISBN: 9780817647728

Category: Mathematics

Page: 338

View: 1492


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 Oxford Handbook of the History of Mathematics

Author: Eleanor Robson,Jacqueline Stedall

Publisher: OUP Oxford

ISBN: 0191607444

Category: Mathematics

Page: 926

View: 989


This Handbook explores the history of mathematics under a series of themes which raise new questions about what mathematics has been and what it has meant to practise it. It addresses questions of who creates mathematics, who uses it, and how. A broader understanding of mathematical practitioners naturally leads to a new appreciation of what counts as a historical source. Material and oral evidence is drawn upon as well as an unusual array of textual sources. Further, the ways in which people have chosen to express themselves are as historically meaningful as the contents of the mathematics they have produced. Mathematics is not a fixed and unchanging entity. New questions, contexts, and applications all influence what counts as productive ways of thinking. Because the history of mathematics should interact constructively with other ways of studying the past, the contributors to this book come from a diverse range of intellectual backgrounds in anthropology, archaeology, art history, philosophy, and literature, as well as history of mathematics more traditionally understood. The thirty-six self-contained, multifaceted chapters, each written by a specialist, are arranged under three main headings: 'Geographies and Cultures', 'Peoples and Practices', and 'Interactions and Interpretations'. Together they deal with the mathematics of 5000 years, but without privileging the past three centuries, and an impressive range of periods and places with many points of cross-reference between chapters. The key mathematical cultures of North America, Europe, the Middle East, India, and China are all represented here as well as areas which are not often treated in mainstream history of mathematics, such as Russia, the Balkans, Vietnam, and South America. A vital reference for graduates and researchers in mathematics, historians of science, and general historians.

The Scientific Legacy of Poincaré

Author: Éric Charpentier,Etienne Ghys,Annick Lesne

Publisher: American Mathematical Soc.

ISBN: 082184718X

Category: Mathematics

Page: 391

View: 6966


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.

Encyclopedia of Nonlinear Science

Author: Alwyn Scott

Publisher: Routledge

ISBN: 1135455570

Category: Reference

Page: 1096

View: 2325


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.

Deleuze and the History of Mathematics

In Defense of the 'New'

Author: Simon Duffy

Publisher: A&C Black

ISBN: 1441113894

Category: Philosophy

Page: 208

View: 2130


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.

The Three-Body Problem

Author: Catherine Shaw

Publisher: Allison & Busby

ISBN: 074901444X

Category: Fiction

Page: 352

View: 3060


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...

A History in Sum

Author: Steve Nadis

Publisher: Harvard University Press

ISBN: 0674727894

Category: Mathematics

Page: N.A

View: 6612


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

The Evolution of an International Mathematical Research Community, 1800-1945

Author: Karen Hunger Parshall,Adrian Clifford Rice

Publisher: American Mathematical Soc.

ISBN: 9780821896730

Category: Mathematics

Page: 406

View: 1907


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.

Celestial Encounters

The Origins of Chaos and Stability

Author: Florin Diacu,Philip Holmes

Publisher: Princeton University Press

ISBN: 9780691005454

Category: Mathematics

Page: 256

View: 8583


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.

History of Topology

Author: I.M. James

Publisher: Elsevier

ISBN: 9780080534077

Category: Mathematics

Page: 1056

View: 9453


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

Author: Horst Drechsel,Miloslav Zejda

Publisher: Springer Science & Business Media

ISBN: 9781402031311

Category: Science

Page: 488

View: 5210


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.

Henri Poincaré, 1912–2012

Poincaré Seminar 2012

Author: Bertrand Duplantier,Vincent Rivasseau

Publisher: Springer

ISBN: 3034808348

Category: Mathematics

Page: 233

View: 1008


This thirteenth volume of the Poincaré Seminar Series, Henri Poincaré, 1912-2012, is published on the occasion of the centennial of the death of Henri Poincaré in 1912. It presents a scholarly approach to Poincaré’s genius and creativity in mathematical physics and mathematics. Its five articles are also highly pedagogical, as befits their origin in lectures to a broad scientific audience. Highlights include “Poincaré’s Light” by Olivier Darrigol, a leading historian of science, who uses light as a guiding thread through much of Poincaré ’s physics and philosophy, from the application of his superior mathematical skills and the theory of diffraction to his subsequent reflections on the foundations of electromagnetism and the electrodynamics of moving bodies; the authoritative “Poincaré and the Three-Body Problem” by Alain Chenciner, who offers an exquisitely detailed, hundred-page perspective, peppered with vivid excerpts from citations, on the monumental work of Poincaré on this subject, from the famous (King Oscar’s) 1889 memoir to the foundations of the modern theory of chaos in “Les méthodes nouvelles de la mécanique céleste.” A profoundly original and scholarly presentation of the work by Poincaré on probability theory is given by Laurent Mazliak in “Poincaré’s Odds,” from the incidental first appearance of the word “probability” in Poincaré’s famous 1890 theorem of recurrence for dynamical systems, to his later acceptance of the unavoidability of probability calculus in Science, as developed to a great extent by Emile Borel, Poincaré’s main direct disciple; the article by Francois Béguin, “Henri Poincaré and the Uniformization of Riemann Surfaces,” takes us on a fascinating journey through the six successive versions in twenty-six years of the celebrated uniformization theorem, which exemplifies the Master’s distinctive signature in the foundational fusion of mathematics and physics, on which conformal field theory, string theory and quantum gravity so much depend nowadays; the final chapter, “Harmony and Chaos, On the Figure of Henri Poincaré” by the filmmaker Philippe Worms, describes the homonymous poetical film in which eminent scientists, through mathematical scenes and physical experiments, display their emotional relationship to the often elusive scientific truth and universal “harmony and chaos” in Poincaré’s legacy. This book will be of broad general interest to physicists, mathematicians, philosophers of science and historians.

Combinatorial Homotopy and 4-Dimensional Complexes

Author: Hans-Joachim Baues

Publisher: Walter de Gruyter

ISBN: 3110854481

Category: Mathematics

Page: 407

View: 3074


The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

Philosophy of Science

The Historical Background

Author: Joseph J. Kockelmans

Publisher: Transaction Publishers

ISBN: 9781412830829

Category: Philosophy

Page: 496

View: 6396


This anthology of selections from the works of noted philosophers affords the student an immediate contact with the unique historical background of the philosophy of science. The selections, many of which have not been readily accessible, follow the development of the philosophy of science from 1786 to 1927. Each selection is preceded by a brief introduction by the editor designed to familiarize the reader with a particular philosopher and provide insights into his work. Joseph J. Kockelmans divides the selections into several sections. Part 1, from 17861850, includes chapters by Immanuel Kant, on the metaphysical foundations of natural science, John Frederick William Herschel, on experience and the analysis of phenomena, William Whewell, on the nature and conditions of inductive science, and John Stuart Mill, on induction and the law of universal causation; part 2, from 18701899, includes chapters by Hermann Von Helmholtz, on the origin and significance of geometrical axioms, William Stanley Jevons, on the philosophy of inductive inference, John Bernard Stallo, on the kinetic theory of gasses and the conditions of the validity of scientific hypotheses, Ernst Mach, on the economical nature of physical inquiry, Karl Pearson, on perceptual and conceptual space, Emile Boutroux, on mechanical laws, Heinrich Hertz, on the appropriateness, correctness, and permissibility of scientific theories, and Ludwig Boltzmann, on the fundamental principles and basic equations of mechanics. The third part, covering the first decade of the twentieth century, includes chapters by Henri Jules Poincare, on science and reality, Charles Peirce, on Induction, Pierre Marie Duhem, on the laws of physics, William Ostwald, on energetism and mechanics, Emile Meyerson, on identity of thought and nature as the final goal of science, Ernst Cassirer, on functional concepts of natural science; part 4, from 19101927, includes chapters by Charles Dunbar Broad, on phenomenalism, Alfred North Whitehead, on time, space, and material, Bertrand Russell, on the world of physics and the world of sense, Norman Robert Cambbell, on the meaning of science, Moritz Schlick, on basic issues of the philosophy of natural science, and Percy Williams Bridgman, on the concepts of space, time, and causality. Philosophy of Science provides a concise single volume text to the discipline and enables students to understand and evaluate the various trends in our contemporary philosophy of science. Joseph J. Kockelmans is professor emeritus of philosophy at the Pennsylvania State Univers

The Legacy of John Von Neumann

Author: James G. Glimm,John Impagliazzo,Isadore Singer

Publisher: American Mathematical Soc.

ISBN: 9780821868164

Category: Mathematics

Page: 334

View: 8688


The ideas of John von Neumann have had a profound influence on modern mathematics and science. One of the great thinkers of our century, von Neumann initiated major branches of mathematics--from operator algebras to game theory to scientific computing--and had a fundamental impact on such areas as self-adjoint operators, ergodic theory and the foundations of quantum mechanics, and numerical analysis and the design of the modern computer. This volume contains the proceedings of an AMS Symposium in Pure Mathematics, held at Hofstra University, in May 1988. The symposium brought together some of the foremost researchers in the wide range of areas in which von Neumann worked. These articles illustrate the sweep of von Neumann's ideas and thinking and document their influence on contemporary mathematics. In addition, some of those who knew von Neumann when he was alive have presented here personal reminiscences about him. This book is directed to those interested in operator theory, game theory, ergodic theory, and scientific computing, as well as to historians of mathematics and others having an interest in the contemporary history of the mathematical sciences. This book will give readers an appreciation for the workings of the mind of one of the mathematical giants of our time.

Turbulence in Economics

An Evolutionary Appraisal of Cycles and Complexity in Historical Processes

Author: Francisco Louçã

Publisher: Edward Elgar Publishing

ISBN: 9781782543671

Category: Business & Economics

Page: 383

View: 1149


'It is difficult to summarize in a short space the extreme richness of this book, which involves arguments taken from physics, philosophy, history of science and epistemology, as well as economic thought and recent developments in econometrics. . . . Louçã's book makes for extremely interesting and useful reading: it provides a solid criticism of the foundations of neoclassical theory and constitutes the unavoidable starting point for any theoretical construction aiming to understand real societies. . . . The vast erudition of the author - who moves easily in many fields of the social and natural sciences - makes the book a mine of information and a valuable source of new ideas.' - Angelo Reati, Review of Political Economy 'This book will be a landmark in the history of economic thought. It is an extremely powerful and original critique of mainstream econometrics, based on a thorough knowledge of its historical origins and its contemporary applications. It will be essential reading for everyone involved in teaching or learning economic theory and model-building. The book also provides new insights into the work of Frisch, Keynes and Schumpeter . . . it is also a very important contribution to philosophy in the social sciences and in particular, to the development of evolutionary theory in economics. The rapid recent growth of interest in evolutionary theory means that the book will be of special interest to those concerned with these exciting new developments.' - Christopher Freeman, SPRU - Science and Technology Policy Research, University of Sussex, UK and Maastricht University, The Netherlands