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Über dieses Buch

The book describes the life of Henri Poincaré, his work style and in detail most of his unique achievements in mathematics and physics. Apart from biographical details, attention is given to Poincaré's contributions to automorphic functions, differential equations and dynamical systems, celestial mechanics, mathematical physics in particular the theory of the electron and relativity, topology (analysis situs). A chapter on philosophy explains Poincaré's conventionalism in mathematics and his view of conventionalism in physics; the latter has a very different character. In the foundations of mathematics his position is between intuitionism and axiomatics.

One of the purposes of the book is to show how Poincaré reached his fundamentally new results in many different fields, how he thought and how one should read him. One of the new aspects is the description of two large fields of his attention: dynamical systems as presented in his book on `new methods for celestial mechanics' and his theoretical physics papers. At the same time it will be made clear how analysis and geometry are intertwined in Poincaré's thinking and work.In dynamical systems this becomes clear in his description of invariant manifolds, his association of differential equation flow with mappings and his fixed points theory.

There is no comparable book on Poincaré, presenting such a relatively complete vision of his life and achievements. There exist some older biographies in the French language, but they pay only restricted attention to his actual work. The reader can obtain from this book many insights in the working of a very original mind while at the same time learning about fundamental results for modern science



The Life of Henri Poincare

Chapter 1. The Early Years

Jules-Henri Poincaré, called Henri, was born in 1854, in Nancy, the capital of the duchy of Lorraine, which in 1766 had become part of France. He died in 1912, in Paris, 58 years old, from a complication following an operation.
Ferdinand Verhulst

Chapter 2. Academic Education: 1873–1879

In the fall of 1873, Henri, then 19 years old, travelled to Paris, accompanied by his mother and sister, to enroll in the École Polytechnique. While in Paris, Henri’s mother and Aline stayed with the Rinck family, old friends from Lorraine, whose son, Élie Rinck, was of the same age as Henri. They remained in Paris a week, during which time they visited Henri and saw him for the first time in uniform (see Figure 2.1); they found it very difficult to say goodbye. The feeling was mutual, attested by the fact that during his first two years in Paris, Henri wrote hundreds of letters home, more than two a week.
Ferdinand Verhulst

Chapter 3. Impressive Results in Vesoul and Caen

In examining the lives of creative people, including scientists and artists, one frequently observes an initial period of acquisition of knowledge and practical skills followed by a burst of activity with occasional interruptions. For Henri Poincaré, this watershed came around 1878. In his case, however, the enormous flow of significant results continued uninterrupted throughout his life.
Ferdinand Verhulst

Chapter 4. Career in Paris

The marriage of Henri Poincaré and Louise Poulain d’Andecy took place on April 20, 1881, in Paris. Their first home in Paris was at Rue Gay-Lussac 66. Later, they moved to Rue Claude Bernard 63. In 1887, their first child was born, a daughter, Jeanne; two daughters were born a few years later, Yvonne in 1889 and Henriette in 1891. The year 1893 saw the birth of their fourth and last child, a son, Léon.
Ferdinand Verhulst

Chapter 5. The Prize Competition of Oscar II

In the summer of 1885, an announcement of a prize competition appeared in several scientific journals. The announcement, from Gösta Mittag-Leffler, stated that King Oscar II of Sweden and Norway had decided to sponsor a scientific competition, with a prize to be awarded on January 21, 1889, his 60th birthday. The practical aspects of the competition were the responsibility of three committee members: chairman Gösta Mittag-Leffler (Stockholm), Karl Weierstrass (Berlin), and Charles Hermite (Paris). The prize would consist of a gold medal and the sum of 2500 kronor. The memoirs offered for the competition were to be submitted by June 1, 1888.
Ferdinand Verhulst

Chapter 6. Philosophy and Essays

A number of Henri Poincaré’s essays are usually classified as “philosophical.” Most of them have been collected in six books. The first five books do not mention the provenance of the original papers, which appeared in various periodicals, such as Revue de Métaphysique et de Morale, often with mathematical details and references that were left out in the book versions. The omission of sources made the books accessible to a wide public, and there were probably also marketing considerations, since the absence of sources suggested a greater originality of the writings. The last collection, Scientific Opportunism, published in 2002, makes up for this omission by including a list of sources and a description of the background of the five books that appeared in the years up to 1913, which were published by Ernest Flammarion, whose brother Camille was an amateur astronomer and prolific popularizer.
Ferdinand Verhulst

Chapter 7. At the End, What Kind of a Man?

Henri Poincaré died on July 17, 1912. During his last year he had been very active, travelling to international conferences and fulfilling many other obligations. On June 26, a few weeks before he died, he gave the public lecture on moral education presented in Chapter 12. The prostate problems that he had experienced at the 1908 international conference in Rome became more serious, and he was advised to have an operation. On Saturday, July 6, he was at a faculty meeting discussing group theory; after that meeting, he told his friend Paul Appell, “Tomorrow I will enter the hospital.” The operation took place on July 9 and seemed to have been successful; family members and friends rejoiced and were reassured. A week later an embolism suddenly terminated his life.
Ferdinand Verhulst

Scientific Details and Documents


Chapter 8. Automorphic Functions

The theory of automorphic functions, or Fuchsian functions as Poincaré called them, is a fruitful result of using complex function theory in the analysis of linear ordinary differential equations (ODEs). The early history of its development has been described in.
Ferdinand Verhulst

Chapter 9. Differential Equations and Dynamical Systems

Henri Poincaré presented his thesis to the Faculté des Sciences of the University of Paris to obtain the degree of doctor of mathematical sciences. The title: “Sur les propriétés des fonctions définies par les équations aux différences partielles.” It was accepted on August 1, 1879, by a committee consisting of J.-C. Bouquet (chairman), P.-O. Bonnet, and G. Darboux. The text is reproduced in [Poincaré 1916, Vol. 1].
Ferdinand Verhulst

Chapter 10. Analysis Situs

In the middle of the nineteenth century, Michel Chasles strongly advocated that geometry and analysis be considered complementary disciplines, not to be separated if one wanted a complete picture of a mathematical theory. To put it simply, analysis provides shortcuts and routine in proofs, while geometry gives insight, showing the meaning of the results. Henri Poincaré’s dissertation advisor, Gaston Darboux, was a student of Chasles. Poincaré was 12 years younger than Darboux, but he underwent a similar influence by studying the writings of Chasles. It shows in his early treatment of ODEs, where he introduced the geometry of the flow near critical points (equilibria) and used projection methods to clarify the structure of solution space. His geometric ideas helped him to handle automorphic functions, where he proposed the relationship between singularities of linear differential equations, Riemann sheets, and non-Euclidean geometry. That influence also came out abundantly in his analysis of dynamical systems, including conservative systems. His concept of “consequents” (Poincaré map) led him to formulate fixed-point theorems to obtain periodic solutions; see Section 9.5. The dynamics of high-dimensional dynamical systems with homoclinic and heteroclinic solutions together with their doubly asymptotic manifolds required subtle analysis in combination with geometric visualization. Also, in his work on the Laplace and Poisson equations, Poincaré’s balayage method clearly pictures the analytic tool of shifting mass distributions in a convenient way.
Ferdinand Verhulst

Chapter 11. Mathematical Physics

In this chapter, we will first look at new methods developed for partial differential equations, and in the following subsections, at a number of applications and physical theories. We aim at conveying the ideas while leaving technical details to the literature cited. We will leave out dynamical systems, since they were discussed in a separate chapter.
Ferdinand Verhulst

Chapter 12. Poincaré’s Address to the Society for Moral Education

Nowadays, scientists are not too fond of “moral societies.” Moralizing as such tends to arouse suspicion, for it often takes place for reasons that are not mentioned explicitly. Also, societies with a moralizing agenda are often sponsored by wealthy business executives, a class with which many intellectuals do not want to be associated.
Ferdinand Verhulst

Chapter 13. Historical Data and Biographical Details

We summarize here some data and events that are characteristic of the social and political climate in which the Poincaré family lived: The Duchy of Lorraine was incorporated into France in 1766. Part of Alsace, the Republic of Mulhouse, became a region of France in 1798.
Ferdinand Verhulst


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