# Jean François Chazy

### Quick Info

Born
15 August 1882
Villefranche-sur-Saône, France
Died
9 March 1955
Paris, France

Summary
Jean Chazy was a French mathematician and astronomer who worked on the three body problem.

### Biography

Jean Chazy's father was an industrialist, working in the textile trade in Villefranche-sur-Saône, a town in east-central France with metallurgical, textile and chemical industries. Jean began his studies at the Collège of Mâcon and then went on to perform outstandingly at the Lycée in Dijon. At this stage in his career he made the decision not to follow the career of his father, but rather to aim for a career in teaching and research. After becoming a Concours Général laureate, he took the entrance examinations of both the École Polytechnique and the École Normale Supérieure, being offered a place by both famous grandes écoles. At this time the École Normale Supérieure had the best reputation for mathematics and, as this was Chazy's chosen topic, he began his studies at the École normale. He graduated with his first degree in 1905.

France's defeat in the Franco-Prussian war of 1870-71 meant that the French were determined to recover Alsace-Lorraine and so were continually preparing for conflict with Germany. Conscription was one way of achieving this, and after graduating, Chazy undertook military service for a year before continuing with his studies. The research he undertook for his doctorate involved the study of differential equations, in particular looking at the methods used by Paul Painlevé to solve differential equations that Henri Poincaré and Émile Picard had failed to solve. Although Painlevé had made many advances, he had also posed many questions and it was these that Chazy attacked. Chazy published several short papers while undertaking research, for instance Sur les équations différentielles dont l'intégrale générale est uniforme et admet des singularités essentielles mobiles (1909), Sur les équations différentielles dont l'intégrale générale possède une coupure essentielle mobile (1910) and Sur une équations différentielle du premier ordre et du premier degré (1911). He was awarded his doctorate in 1911 for his thesis Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes which he defended at the Sorbonne on 22 December 1910. It was published as a 68-page paper in Acta Mathematica in 1911. In his thesis he was able to extend results obtained by Painlevé for differential equations of degree two to equations of degree three and higher. It was work of the highest quality.

In 1911, Chazy was appointed as a lecturer in mechanics at the University of Grenoble. By good fortune the Academy of Sciences posed a topic for the Grand Prix in Mathematical Sciences of 1912 which was exactly right given the research that Chazy had undertaken for his thesis. The topic posed was: Improve the theory of differential algebraic equations of the second order and third order whose general integral is uniform. The judges, one of whom was Paul Painlevé, received a number of outstanding submissions for the prize, including one from Chazy. In the end they decided to split the award three ways giving a one-third share to each of Chazy, Pierre Boutroux and René Garnier. Chazy moved from Grenoble to Lille where he was appointed as a lecturer in the Faculty of Sciences (later renamed the University of Lille). However, the outbreak of World War I interrupted his career. By the end of July 1914 France had began mobilizing its troops and, on 3 August, Germany declared war on France. Chazy was mobilised and sent to the sound reconnaissance laboratory which was set up at the École Normale Supérieure. This laboratory was under the direction of Jacques Duclaux who was married to the radiologist Germaine Berthe Appell, making Duclaux the son-in-law of Paul Appell and the brother-in-law of Émile Borel. The Germans had developed a new gun, known as "Big Bertha", a large howitzer built by the armaments manufacturer Krupp. At the start of the war, only two of these guns were operational but they were very effective in taking the Belgium forts at Liège, Namur and Antwerp as well as the French fort at Maubeuge. In April 1918 the German armies made a number of attacks against allied positions which met with considerable success. They swept forward and the French government made strenuous efforts to defend Paris at all costs. On 15 July the French withstood the last German offensive. Chazy was able to compute the position of the Big Bertha guns firing at Paris from long-range with surprising accuracy. This played an important part in the ability of the French to defend the city. On 8 August the German armies began to move back; this was the beginning of the final chapter in the war. For his war work Chazy was awarded the Croix de Guerre.

Once he was released from military service in 1919, Chazy returned to his position in Lille. In addition to his university position, he also took on another teaching position in the city, namely at the Institut industriel du Nord. Lille had been occupied by the Germans between 13 October 1914 and 17 October 1918, during which time the university suffered looting and requisitions. Much damage had been caused, particularly in 1916 when explosions had destroyed the university's laboratories. After the war ended the reconstruction of educational facilities began. It was a difficult time for all those working there. Having done brilliant work on differential equations, Chazy's interests now turned towards the theory of relativity. Albert Einstein's general theory of relativity was, at this time, very new only having been published near the end of 1915. Chazy's first publication on the subject was Sur les fonctions arbitraires figurant dans le $ds^{2}$ de la gravitation einsteinienne which appeared in 1921. In the same year, Chazy published a paper on the three-body problem, Sur les solutions isosceles du Problème des Trois Corps , an area he had begun to study in 1919 and for which he has become famous. Several important papers on the three-body problem followed, such as Sur l'allure du mouvement dans le problème des trois corps quand le temps croît indéfiniment (1922) which led to Chazy being awarded the Prix Benjamin Valz. An important paper on relativity he published in 1924 is Sur le champ de gravitation de deux masses fixes dans la théorie de la relativité .

In 1923 Chazy was appointed as a lecturer at the École Centrale des Arts et Manufactures in Paris and he was also appointed as an examiner at the École Polytechnique. He was named professor of analytical mechanics at the Sorbonne in 1925. He continued to work at the Sorbone until his retirement in 1953, but over the years he worked in Paris he held a number of different chairs such as analytical and celestial mechanics, and rational mechanics. Donald Saari explained some of Chazy's major contributions in [10]:-
... in the 1920s, the French mathematician Jean Chazy made an important advance. While he did not solve the three-body problem in the standard way of explaining what happens at each instant of time, with a clever use of the triangle inequality he was able to discover the asymptotic manner in which these particles separate from one another as time marches on to infinity. His description carefully catalogues all possible ways in which particles can separate from one another, or from a binary cluster. But a side issue arose: was his beautiful and orderly description marred by the inclusion of another, possibly spurious kind of motion? In this regime, the radius of the universe would expand, then contract, then expand even more, then ... All of this oscillating would create a setting whereby, as time goes to infinity, the limit superior of the radius of this universe approaches infinity while the limit inferior is bounded above by some positive constant. Weird, but if this oscillatory behaviour could occur, it would be fascinating. Chazy, however, had no clue as to whether it could exist. He was forced to include the motion only because he could not develop a mathematical reason to exclude it.
Chazy published a number of influential texts while working as a professor in Paris. He published the two-volume treatise The Theory of Relativity and Celestial Mechanics (1928, 1930). He wrote in the Preface:-
The purpose of this book, which is the development of a course taught at the Faculty of Sciences of Paris in 1927, is to expose as clearly as possible the theory of relativity in dealing with celestial mechanics, taking as a starting point the knowledge of a student who has attended a few lessons on differential and integral calculus, and mechanics.
The publisher gave this description of the second volume:-
This second part of the beautiful treatise of Jean Chazy is second to none as the leading report of interest. It discusses the principles of relativity, the equations of gravitation, the determination of $ds^{2}$, Schwarzschild equations of motion, the $n$-body problem and finally cosmogonic hypotheses related to the $ds^{2}$ of the universe. The beautiful clarity in the exposition, the scope of information and the contributions from the author himself make this book indispensable to anyone who wants to enter the heart of the theories to which Einstein's name is attached.
Here are the headings of the twelve chapters in the text.

Volume I:
(1) The calculus of variations. Geodesics of a $ds^{2}$;
(2) The law of gravitation derived from the Schwarzschild $ds^{2}$ and the advance of planetary perihelia;
(3) The law of gravitation of the theory of relativity and the classical theory of perturbations;
(4) The work of Le Verrier and Newcomb;
(5) Explanations of three disagreements between Newtonian theory for the major planets and observation;
(6) The bending of light rays near the Sun.
Volume II:
(7) Birth of the Theory of Relativity. Displacement of spectral lines;
(8) The ten differential equations of gravitation;
(9) The structure of the Schwarzschild $ds^{2}$;
(10) The Laplace equation and the Poisson equation. The velocity of the propagation of gravity. Approximate equations of motion;
(11) Rotation of a central body. The $n$-body problem. Motion of the Moon;
(12) Cosmological hypotheses.
In 1933, Chazy published another two-volume work, namely Cours de Mecanique Rationelle . This work was very popular and he published further editions, the third being published in 1947 (Volume I) and 1948 (Volume II). In a review of the two volumes Philip Franklin writes:-
[The first volume] on the dynamics of a material particle begins with a discussion of vectors. The principles of mechanics are then formulated and in successive chapters applied to problems of equilibrium, motion in one, two and three dimensions, and later to motion on a curve and in a surface. The effects of friction and the rotation of the earth are discussed. The treatment is clear and logical with special attention devoted to qualitative results and the precise consideration of singular situations where the simplest existence theorems do not apply. [The second] volume discusses systems of particles, rigid bodies, strings, hydrostatics, hydrodynamics and gravitational potential. The treatment is clear, concise and exact.
S G Hacker [9] writes:-
Professor Chazy's 'Cours' provides another lucid, and in this case refreshing, introduction to theoretical dynamics.
In 1953 Chazy published Mécanique céleste. Equations canoniques et variation des constantes . G Lampariello writes:-
This monograph should be considered as an important complement to the two major treatises on celestial mechanics by Tisserand [Traité de mécanique céleste ] and Poincaré [Méthodes Les nouvelles de la mécanique céleste; Leçons de mécanique céleste ]. The masterly exposition of the fundamental concepts makes a happy reconciliation between classical and relativistic concepts. The author calculates the shift of the perihelion of the planets and the bending of light in the gravitational field of the Sun and compares the theoretical results with the observations.
Pierre Costabel gives the following overview of Chazy's contributions in [1]:-
While penetrating in an original and profound way the field of research opened by the relativity revolution, Chazy nevertheless remained a classically trained mathematician. With solid good sense, he held a modest opinion of himself. The reporters of his election to the Academy were, however, correct to stress how much, in a period of crisis, celestial mechanics needed men like him, who were capable of pushing to its extreme limits the model of mathematical astronomy that originated with Newton. Thus, beyond the lasting insights that Chazy brought to various aspects of the new theories, his example remains particularly interesting for the philosophy of science.
Chazy received many honours for his contributions. He was elected a member of the astronomy section of the Academy of Sciences on 8 February 1937, and became a member of the Bureau des Longitudes in 1952. He was also a member of the Romanian Academy of Sciences and a member of the Belgian Academy of Sciences. In 1934 he was elected president of the Société Mathématique de France, being followed in this position in 1935 by Maurice Fréchet. He was also elected to the Geographical Society of Peru and the Institute of Coimbra in Portugal. When he retired in 1953 he was made a Commander of the Légion d'Honneur.

### References (show)

1. P Costabel, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. M Valtonen and H Karrunen, The Three Body Problem (Cambridge University Press, Cambridge, 2006).
3. K G Atrokhov and V I Gromak, Solution of the Chazy system (Russian), Differ. Uravn. 46 (6) (2010), 777-790.
4. K G Atrokhov and V I Gromak, Solution of the Chazy system, Differ. Equ. 46 (6) (2010), 783-797.
5. S Chakravarty and M J Ablowitz, Parameterizations of the Chazy equation, Stud. Appl. Math. 124 (2) (2010), 105-135.
6. P Chevenard, Jean Chazy (15 août 1892-9 mars 1955), L'Astronomie 69 (1955), 158.
7. G Darmois, Notice sur la vie et les travaux de Jean Chazy (1882-1955), Notices et discours - Institut de France. Académie des sciences 4 (1964), 35-46.
8. G Darmois, Notice sur la vie et les travaux de Jean Chazy (1882-1955) (Palais de l'Institut, Paris, 1957).
9. S G Hacker, Review: Cours de Mecanique Rationelle by Jean Chazy, Amer. Math. Monthly 56 (9) (1945), 646-647.
10. D G Saari, Review: The Three Body Problem by Mauri Valtonen and Hannu Karrunen, Amer. Math. Monthly 115 (7) (2008), 672-676.