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History of a noted Mathematician: Henri Poincare

Introduction

Human beings by nature seek knowledge throughout their life of what is there, has been there and what is to come. History is a key resource in attaining knowledge as it gives previous information regarding things that happened before. The paper focuses on the famous mathematician Henri Poincare, who is celebrated to date.

Biography

Henri Poincare was born of Nancy in the year 1854 on April 29 in Hotel Martigny which was a town mansion before it was converted to a drugstore. His innate abilities were dominant and they manifested themselves in his great achievements. Their family had a lot of learned individuals as seen from his father who was a professor in the Faculty of Medicine and also a neurologist and the grandfather happened to be a pharmacist. His younger sister was the wife of Emile Boutroux, the famous philosopher and they had a son who was also a philosopher and very talented in mathematics also.

 Henri fell victim of diphtheria at a tender age of five but none the less, his childhood portrayed the stories by ancient books that show different stages of development. On many occasions when he was playing with his sisters and cousins he invented many games, and this showed his strong imaginative capabilities. He got exposed to a clever private tutor who was able to nurture him and show him how to use his stupendous memory.

Upon joining high school in Nancy, his capabilities began manifesting themselves more and more, and he received recognition as a first class student. In his last few years here he proved to be a ‘mathematical monster” and this was only just the inception. He managed to get his baccalaureate in letters and science, and this was an achievement for him as he prepared for his entrance examination for the ” Grand Ecoles”. He spent two years preparing for this exam and during this time he became quite popular.

After finishing high school he was ranked as the best student admitted to Ecole Polytechnique and fifth-best to Ecole Normale Superieure, he decided to attend the former, and upon graduation, he emerged second in his class. Henri then proceeded to Ecole des Mines where his love for mathematics led him to cryptology and all these nights have been the inspiration behind his interest in the group theory.

During his time in Sorbonne, he was denied consent to attend lectures and in August 1876 Poincare managed to attain his diploma mathematics from the Faculty of Science in Paris. While at Ecole des Mines he spent the last two years preparing his thesis in mathematics. August 1879 is the date that Henri got to defend his thesis and he did it before a jury encompassing Darboux, Bonnet, and Bouduet at the Faculty of Science. The thesis included some classical results of Bouquet and Briot about singular ordinary differential equations. A very real order from Daboux was about the methods used and results obtained, but he paid not so much attention on the clarity of the methodology used.

In April 1879 Poincare landed a job as a mining engineer in Vesoul (“Henri Poincare: a scientific biography”, 2013). There was a hot pit called Magny where the lives of sixteen were lost to an explosion of a firedamp. He was on leave for the better part of his stay here and he also managed to get a promotion as a member of the Corps des Mines. His career took an academic turn as he went to teach analysis at Caen, Faculty of Science in 1879. His stay here lasted only two years and he later moves Paris in the Faculty of Science and still lectured on analysis.

Henri went up the ranks in a series of events in consecutive years when he got an appointment as the professor of experimental physics and physical mechanics in 1886 and 1885 respectively followed by 1896 when became a professor of celestial mechanics and mathematical astronomy (Sauer,2014). He also went to Ecole Polytechnic where he taught astronomy and later at the Ecole des Postes et Telegraphs to lecture on theoretical electricity and was also a member of the Bureau des Longitudes. He had a good reputation from his students and fellow mathematicians at large.

During his stay in Caen between August 1879 and October 1881, he experienced tremendous changes in life such as the entrance of three daughters and a son in his life after marrying Louise Poulain d’Andency. Charles Hermite was a lecturer at Ecole Polytechnic taught Poincare analysis, he got his reputation from the work he did such as his proving the transcendental character of the number e. He was reigning over French mathematics at that time and was the source of inspiration for Henri and his study of quadratic and ternary forms. He, on the other hand, reacted positively to Poincare’s work although he was not a fan of geometry.   

Contribution to the field of Mathematics

Henri told the story of how he came up with automorphic functions himself. These functions extend both the elliptic functions and trigonometric functions and recover their values through the action of a specific group of homographic substitutions. Curves bind curvilinear figures is responsible for replacing the tesselation of the complex plane which is made by rectangles for elliptic functions.

He played a big role in the three- body problem which was later called the n- body problem, n being the number of orbiting bodies more than two. It involved cracking the solution to the issue of two bodies orbiting the solar system, and this had been a headache to mathematicians from the time of Isaac Newton (Verhulst,2016). The solution of the n- body problem was very necessary yet quite challenging during the end of the 19th century. Oscar II, King of Sweden, announced that a prize would be given to anyone who could give the solution. Although he did not solve the original problem, the prize was later awarded to Poincare.

During his time at Bureau des, Longitudes Henri worked on establishing international time zones (“Henri Poincare: a scientific biography”, 2013). As he worked on this his interest grew in clocks and how they work while moving about space at different speeds and he wanted to synchronize the clocks. Lorentz came up with an auxiliary quantity in 1895 called ”local time” t’= t – vx/c2. He also aimed at explaining the failure of electrical and optical experiments to detect motion which is about the aether by using the hypothesis of length contraction. Henri was a usual interpreter of this theory by Lorentz and from his philosophic nature, he went for the ‘deeper meaning”.

Henri did two papers in 1900 and 1904 named the principle of relativity where he talked about the ”principle of relative motion”. In his discussion of this paper, he said that distinction between a state of rest and state of uniform motion cannot be done by use of a physical experiment. Poincare and Lorentz exchanged letters where they questioned the theories and made points clear about the theory making it better. Poincare claimed that the combination x2 + y2 + z2 – c2t2 is invariant and termed Lorentz transformation to be a rotation in a space that is four dimensional about the origin with the use of four-vectors in their early forms. With his new mechanics in 1907, he clearly showed his disinterest in four-dimensional reformulation.

In 1900, like his predecessors, Poincare found that there is a relation between electromagnetic energy and mass. He tried to figure out whether with the presence of magnetic fields, whether the center of gravity will still move at the same velocity. It is he who noted that action/reaction theory is not applicable to matter alone, but there is momentum in the electric field. From the fact that energy can be converted into other forms, Henri assumed that at each point of space there exists an energy fluid non- electric in nature whereby electric energy has a mass that is directly proportional to energy and can be transformed into something else (Riley,2016). He also managed to discuss two unexplained effects: non- conversation

Poincare also published two monographs ”Lectures on Celestial Mechanics” (1905- 1910) and ”New Methods of Celestial Mechanics” (1892- 1899) that were classical. In these monographs, he managed to utilize the results obtained during research to the issue of motion of three bodies while also he did a detailed study on the behavior of solutions. He also showed that one cannot integrate the three body problem. Poincare grew better and better ideas and he was the base of the ‘chaos theory’ in mathematics and the general theory of dynamical system. He also came up with the bifurcation concept, showing the presence of equilibrium figures of non- ellipsoid figures

Poincare wrote a series of memoirs between 1881and 1882 with the title ”On curves defined by differential equations where he focused on the study of singular points used in the system of differential equations. He introduced a branch of mathematics named ”Qualitative theory of differential equations”. In his work Henri was trying to show that known functions can also be used to solve differential equations, he also investigated trajectories and their nature in the plane as well as formulating a general theory for solving variation equations and integral invariants.

Conclusion

Henri Poincare was a very good scientist and his work was known all over the world due to the contribution it has to the field of science and mathematics. He was quite interested in how his mind worked and ended up studying his habits and giving a talk on the same. The inventions that he made not only belong to the history of science but also to most of the vibrant mathematics of age. Henri suddenly died on July 17, 1912, when he succumbed to embolism after a surgery and his legacy was just but what was left of him.

References

Henri Poincare: a scientific biography. (2013). Choice Reviews Online50(11), 50-6231-50-6231. http://dx.doi.org/10.5860/choice.50-6231

Riley, N. (2016). Henri Poincaré: a biography through the daily papers, by Jean-Marc Ginoux and Christian Gerini. Contemporary Physics57(2), 267-268. http://dx.doi.org/10.1080/00107514.2016.1156747

Sauer, T. (2014). Jeremy Gray. Henri Poincaré: A Scientific Biography . xiii + 392 pp., illus., apps., bibl., index. Princeton, N.J./Oxford: Princeton University Press, 2013. Isis105(1), 229-231. http://dx.doi.org/10.1086/676785

Stillwell, J. (2014). Henri Poincare. A Scientific Biography–A Book Review. Notices Of The American Mathematical Society61(4), 1. http://dx.doi.org/10.1090/noti1101

Verhulst, F. (2016). Henri Poincaré: A Scientific Biography. The European Legacy21(4), 456-458. http://dx.doi.org/10.1080/10848770.2016.1150077

 

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