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Sergei Alekseevich Chaplygin's father, Aleksei Timofeevich Chaplygin, was a shop assistant who died of cholera at the age of 24 when Chaplygin was only two years of age. This meant that he was brought up in somewhat poor circumstances by his mother Anna Petrovna. However, Anna Petrovna remarried a tradesman from Voronezh and the family left Ranenburg and moved to Voronezh. His mother was keen for Sergei Alekseevich to have an education so, when he was eight years old, she hired a tutor to bring his education up to the standard required for taking the entrance examinations to the preparatory classes. In the autumn of 1877 he took these examinations and began his schooling.
As we have indicated, the family were poor so they would have had great difficulty paying Sergei Alekseevich's school fees. However, as soon as he began his schooling it became clear to his teachers that he had exceptional abilities and a remarkable memory. On his teachers' recommendation the Pedagogical Board decided to waive the fee requirement in his case. While attending Voronezh Gymnasium, the parents of his fellow pupils, realising his extraordinary skills, asked him to tutor their sons. He therefore began his teaching career at the early age of fourteen, becoming a popular private tutor. The money he made from private tutoring helped to support his family. He excelled across the whole range of school subjects, being equally good at ancient languages, modern languages, history, and his other subjects. However, it was mathematics that he enjoyed most and he decided that he wanted to make a career with this subject. He graduated from the Gymnasium in 1886 with the gold medal and a prediction by his teacher A P Kiselev that:-
... Sergei Alekseevich has a great future.
In the autumn of that year, at age seventeen, Chaplygin packed a small suitcase and, with 200 rubles in his pocket, set off from Voronezh to Moscow to begin his university studies of mathematics.
At the University of Moscow Chaplygin studied in the Physics and Mathematics Faculty and, for the first two years, he concentrated exclusively on mathematics courses. However, beginning in his third year his interests moved towards specialising in mechanics. He was strongly influenced by Nikolai Egorovich Zhukovsky who lectured to him on a wide range of topics. After Chaplygin graduated with his first degree in 1890, Zhukovsky persuaded him to continue to study for his university teacher's qualification. His first work was on hydrodynamics, done under Zhukovsky's influence; in particular on the mechanics of liquids and gases, studying jet stream flow in the 1890s. In 1893 Chaplygin published On certain cases of the motion of a solid body in a fluid and the quality of this work was so outstanding that he was awarded the N D Brashman Prize for this contribution. In 1897 he submitted his master's thesis (equivalent to a Ph.D.) which was published under the same title in 1893. Grigorian writes that :-
Chaplygin gave a geometric interpretation of those cases of the movement of a body in a liquid that had earlier been studied from a purely analytic standpoint by the German scientists Clebsch and Kirchhoff, as well as by the Russian scientist Steklov. In this regard Zhukovsky has written that Chaplygin "demonstrated in his two excellent papers what strength the cleverly conceived geometrical methods of investigation can possess."
The St Petersburg Academy of Sciences awarded him their Gold Medal for this outstanding work.
Chaplygin had begun teaching in Moscow in the year 1893. He taught physics at the Women's Secondary Educational Institution and mechanics at the Moscow Higher Technical School from 1896 until 1906, and at the Moscow Engineering School. He taught from 1894 at Moscow University as an assistant professor, being promoted to professor in 1903. From 1901 he was professor of mechanics at Moscow Women's College and in 1905 he helped organise the Moscow Advanced Course for Women. He served as director of the Advanced Course for Women from 1905 to 1918 when the Advanced Course became the Second Moscow State University. Keldysh writes that :-
... under his leadership the advanced courses developed into a large higher educational institution offering instruction in all branches of knowledge.
He became rector of this university for one year 1918-19 until the two universities of Moscow were merged. As an indication of Chaplygin's remarkable memory, it is recorded in  that during his time as rector he knew every student by name.
After the award of his Master's Degree (equivalent to a Ph.D.), Chaplygin worked on two classic problems of theoretical mechanics. One of these was the study of the motion of a body subjected to nonintegrable constraints while the other was the study of the motion of a heavy body about a fixed point. His paper On the motion of a heavy body of revolution in a horizontal plane (1897) was the first to present the general equation of motion of a nonholonomic system. This equation is a generalisation of Lagrange's equation. In the same year he published On a certain possible generalisation of the theorem of areas. In 1899 he was awarded the Gold Medal of the St Petersburg Academy of Sciences:-
... for his studies of the theory of the motion of a body in a fluid and the motion of bodies with nonintegrable constraints.
He published a famous paper On gas streams in 1902 giving exact solutions to many cases of noncontinuous flow of a compressible gas. This paper, which he submitted to Moscow University for his doctorate (equivalent to the habilitation or D.Sc.), opened the way for a study of high velocity aeromechanics as Keldysh explains :-
... it gave a method of studying jet flows of a gas at any subsonic speed. At that time the investigation of gas flows at speeds near the speed of sound had no practical application in aviation. Three decades later, however, Chaplygin's dissertation served as a starting point for many studies by aerodynamics specialists and provided the basis for the solution of problems of subsonic flows.
In fact Chaplygin wrote his remarkable dissertation while he was on vacation at a resort in the Crimea.
The political situation in Russia led to problems in Moscow University that had major consequences for Chaplygin. The University had been founded in 1863 but after a few years of academic freedom, it was restricted in its activities. Nevertheless it continued to operate as an excellent university, providing Chaplygin with a very good education. However, the first Russian Revolution of 1905-07 saw the students organise themselves into a group seeking the overthrow of the tsar. After this First revolution failed, the tsar took action against Moscow University and it was closed on a number of occasions. Troops were stationed on the campus and action was taken against professors who the tsar thought had been behind the anti-tsarist movement. In 1911 around 130 professors resigned in protest at the actions that were being taken against the university and its professors. Chaplygin was one of the professors who resigned in 1911 along with a number of his colleagues in mathematics such as Boleslav Kornelievich Mlodzeevskii (1858-1923), who was very active in the Moscow Mathematical Society.
In fact it was the Moscow Mathematical Society which played an important role in keeping Moscow mathematicians in contact with each other during the years after they resigned their university positions. Zhukovsky was the president of the Moscow Mathematical Society at this time which gave Chaplygin more reason to maintain a high level of involvement. Chaplygin read papers on mechanics and on approximate methods in analysis to the Society. He had developed methods of approximation for solving differential equation and he presented his first results on that topic to the Moscow Mathematical Society in 1905. From 1910 he studied the theory of the aeroplane wing, presenting a paper to the Society in February of that year which determined the flow around a wing section. In the same year he published his results in the paper On the pressure exerted by a plane-parallel flow on an obstructing body, which gave the lift on a wing section :-
The postulate concerning the determination of the rate of circulation around a wing was first precisely stated in this paper. This postulate - the so-called Chaplygin-Zhukovsky postulate - gives a complete solution to the problem of the forces exerted by a stream on a body passing through it. This article includes the fundamentals of plane aerodynamics, particularly Chaplygin's celebrated formulas for calculating the pressures exerted by the stream of a fluid on an impeding body. These formulas were applied by Chaplygin to the calculation of the stream pressure on various wing profiles for which he gives the construction.
Khristianovich writes about this work in :-
The most notable feature of this study is the statement that if the wing profile has a rounded leading edge and a sharp trailing edge then as it moves uniformly in a fluid, a circulatory flow is established with continuously shedding flow at the sharp trailing edge. In this case a lift proportional to the angle of attack develops.
Chaplygin's important paper Theory of cascaded airfoils (1914) presented the basic theory of circulation round cascades, the theory which is used in the design of propellers, turbines and other hydraulic devices.
Together with Zhukovsky, Chaplygin planned for the setting up of an aeronautical research centre in Moscow. Following the Russian October Revolution of 1917, Lenin agreed that such a centre should be set up. The Central Aerohydrodynamic Institute, or TsAGI, was founded in 1918 and Chaplygin helped organise the Institute from that time. In fact from the time that TsAGI was founded, he devoted almost all his energies to the project. On the death of Zhukovsky in 1921, Chaplygin became Chairman of the Board, a position he held until 1930. He was Executive Director of the Institute from 1928 to 1931, then he became head of the scientific work of the Institute. Under his leadership excellent laboratories were built at TsAGI, including a wind tunnel in 1925.
Lazar Aronovich Lyusternik became a research student in the Moscow Institute of Mathematics and Mechanics in 1922 where he was taught by Chaplygin. He describes in  the extremely high standards that Chaplygin had for the top students:-
In my student days, he used to give courses on the mechanics of a particle and of a system, but even at the lectures of this most distinguished scholar there was sometimes only one student present - Minakov (later professor of mechanics), whom Chaplygin used to 'torture', so the story goes, by making him solve difficult mechanics problems.
Lyusternik also describes Chaplygin's appearance:-
Chaplygin had a distinctive appearance, as if he was made up entirely of angles. He would have been a good model for the then fashionable cubist portraits. I think some portraits that soften this angularity deprive Chaplygin of his individuality.
However, there were political problems in the Soviet Union during the 1930s which made the life of many top scientists extremely difficult. On 3 July 1936, Nikolai Nikolaevich Luzin, Professor of Pure Mathematics at Moscow University, was the victim of a violent political campaign organized by the Soviet authorities through the newspaper Pravda. In the article Enemies under the Mask of a Soviet Citizen he was accused of anti-Soviet propaganda and sabotage by publishing all his important results abroad and only minor papers in Soviet journals. The article claimed that Luzin combined:-
... moral unscrupulousness and scientific dishonesty with deeply concealed enmity and hatred to every bit of the Soviet life.
A week after the article appeared, Chaplygin wrote to V I Vernadsky explaining how he felt at the attack on his colleague:-
The article about Luzin is completely outrageous: Supposing that he committed the sin of misjudging some applicant for a scientific degree or title, but how is it possible to jump to the conclusion of sabotage from that?! ... As far as the accusations, slipped in throughout the article, of fascism and his enlisting the old reactionary Moscow school of mathematics, I am complete unable to understand these. There remains the critical evaluation of Luzin's contributions. But in this regard I must say only that this discloses the complete incompetence of the authors which proves their minor and superficial acquaintance with his works and their deliberate distortion of correct evaluation. His authority is incomparable with that of Khinchin who is counterpoised to him. But what should be done right away? How can we help N. N.? I did only one thing so far: I sent N. N. a cable whose copy I attach: "Dumbfounded by absolutely undeserved newspaper attacks against you. Your high world-wide acknowledged scientific authority cannot be shaken. I hope definitely that you will find the inner forces to face this inauthoritative criticism of your contributions calmly. I avoid mentioning the completely groundless accusations of the other sort."
Chaplygin received many awards for his outstanding contributions. He was elected a corresponding member of the USSR Academy of Sciences on 6 December 1924 then, four years later, he became a full member on 12 January 1929. Also in 1929 he received the title of Honoured Scientist. He was awarded the N E Zhukovsky Prize in 1925. He received two Orders of Lenin, two Orders of the Red Banner of Labour, and was awarded the title Hero of Socialist Labour in 1941. As an indication of Chaplygin's way of thinking, we quote a story told by Lyusternik :-
One day Sergei Alekseevich pulled from some bag of old papers a manuscript containing the foundations of a theory of the boundary layer, developed as far as Prandtl's work. But Chaplygin had not published this paper, since this theory did not have a mathematical basis.
Although World War II started in September 1939 with the German and Russian invasion of Poland, this had little effect on the Central Aerohydrodynamic Institute in Moscow or on its staff. However, on 22 June 1941 the German armies invaded their former ally pushing rapidly east into Soviet lands. They advanced along a broad front from the Baltic to the Black Sea and, in the autumn of 1941, the Central Aerohydrodynamic Institute was evacuated from Moscow to Kazan and Novosibirsk. Chaplygin took charge of the Novosibirsk branch of the Institute and rapidly organised the building of a wind tunnel and research laboratories. However, the hard work and difficult circumstances told on his health and he died from a brain haemorrhage in October 1942.
Further honours were given to Chaplygin following his death. In 1942 the USSR Academy of Sciences established the S A Chaplygin Prize:-
For the best original work in theoretical research in the field of mechanics.
Ranenburg, the town of his birth, was renamed Chaplygin in 1948. His collected papers were published in 1948-1950 in four volumes: Volume 1. Theoretical mechanics and mathematics; Volume 2. Hydromechanics and aerodynamics; Volume 3. Lectures and presentations on mathematics and mechanics; and Volume 4. Lecture courses on theoretical mechanics. Selections of his many papers on mechanics and mathematics were re-published in 1954 and 1976. A crater on the moon has been named for Chaplygin.
Finally let us mention that Chaplygin loved playing chess and spent many hours playing against friends in the staff club.
Article by: J J O'Connor and E F Robertson
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