Aleksandr Mikhailovich began his education at home, then later one of his uncles R M Sechenov prepared him for entering the Gymnasium. Lyapunov was not the only one being coached by Sechenov who was teaching his own daughter Natalia Rafailovna Sechenov at the same time. In fact Natalia and Aleksandr married many years later when he was 29 years old. Some years after the death of Lyapunov's father, Sofia Aleksandrovna moved to Nizhny Novgorod (named Gorky from 1932 to 1990) in 1870 with her children and Lyapunov entered the Gymnasium in that city. There he was a school friend of Markov. He graduated in 1876 and, like his friend Markov, entered the Faculty of Physics and Mathematics at St Petersburg University.
At St Petersburg University he was taught by Chebyshev who, as we shall see below, had a strong influence on him. Lyapunov graduated in 1880 and remained at St Petersburg to undertake research. He published two papers on hydrostatics in 1881: On the equilibrium of heavy bodies in heavy liquids contained in a vessel of a certain shape, and On the potential of hydrostatic pressures. In the following year Chebyshev posed a question to Lyapunov which would set the agenda for one of his main lines of research over many years:-
It is known that at a certain angular velocity ellipsoidal forms cease to be the forms of equilibrium of a rotating liquid. In this case, do they not shift into some new forms of equilibrium which differ little from ellipsoids for small increases in the angular velocity?Although Lyapunov's Master's thesis did not answer this question, the work of the thesis was motivated by it. He presented the thesis On the stability of ellipsoidal forms of equilibrium of a rotating liquid in 1884 and defended it at St Petersburg University in the following year. Following this he was appointed as a privatdozent at Kharkov University where he taught mechanics and continued research for his doctoral thesis. He presented his doctoral thesis The general problem of the stability of motion to the University of Moscow and was awarded his doctorate after defending the thesis on 12 October 1892 (according to the modern calendar). The importance of this thesis is emphasised in several articles such as , ,  which were all written to celebrate the centenary of the publication of this fundamental contribution.
In the following year he was appointed as a professor at Kharkov University where he remained until 1902. While at Kharkov University he played a major role in the Kharkov Mathematical Society, being its vice-president from 1891 to 1898 and president from 1899 until he left Kharkov in 1902. He also edited the Communications of the Kharkov Mathematical Society.
In  Pavlovskaya looks at Lyapunov's work on the problem first posed by Chebyshev which we quoted above. The problem posed by Chebyshev concerning the existence of figures of equilibrium, in addition to ellipsoidal ones, of a rotating fluid under sufficiently small variations of angular velocity of revolution was first solved by Lyapunov in a first approximation. He later dealt with the problem of stability of fluid ellipsoids basing his investigations on the Thomson-Tait variational principle. He showed that a sufficient condition for stability is that the second and higher variations of the potential energy are positive. Lyapunov admitted that the imposition of certain additional constraints on the first variation reduced the generality of his method, but writes:-
But in this respect hardly any other method of investigation could be said to be completely satisfactory.Lyapunov established that with variation in the angular velocity of revolution Maclaurin ellipsoids pass into Jacobi ellipsoids. The transition point is an ellipsoid of bifurcation corresponding in this case to a Jacobi ellipsoid of revolution.
In 1901 Lyapunov was elected to the Russian Academy of Sciences in St Petersburg and in the following year became an academician in applied mathematics of the Academy. Grigorian writes :-
In St Petersburg, Lyapunov devoted himself completely to scientific work. He returned to the problem that Chebyshev had placed before him and, in an extensive series of papers which continued until his death, developed the theory of figures of equilibrium of rotating heavy liquids.In 1917 Lyapunov left St Petersburg to take up a post at the university in Odessa, on the Black Sea coast. He taught at the university but in the spring of 1918 his wife's health began to deteriorate rapidly. Natalia Rafailovna suffered from a form of tuberculosis and Lyapunov was greatly disturbed to watch her health fail. On 31 October 1918 Lyapunov's wife died and later that day Lyapunov shot himself. He died three days later in hospital.
We have described Lyapunov's main work which was on the theory of rotating liquids. There are, however, other aspects of his work we should mention. One is certainly his contributions to probability which he became interested in because of courses he was teaching on that subject. In particular in two papers published in 1900 and 1901, he proved the central limit theorem using a technique based on characteristic functions. Another contribution which we should mention is that as editor for two volumes of Euler's collected works.
He was honoured for his outstanding contributions by election to various academies such as the Accademia dei Lincei (1909) and the French Academy of Sciences (1916). He was also given honorary membership of the universities of St Petersburg, Kharkov and Kazan. Various tributes were paid to him on the centenary of his birth. For example on 6 June 1957 Sobolev gave the lecture On the works of A M Lyapunov on potential theory in Moscow to a joint session of the Presidium of the Academy of Sciences, the divisions of technical and physical sciences of the Academy of Sciences, the Moscow University, the Moscow Mathematical Society, the Institute of Mechanics of the Academy of Sciences, and the Institute of Automatics and Telemechanics of the Academy of Sciences. The text of this lecture is given in .
Other papers such as  describe the way that Lyapunov's contributions to the stability of motion has influenced the development of the subject over a long period of time. Topics considered in  include: stability, particularly the stability of critical points; the construction and the application of the Lyapunov function; stability of functional- differential equations; the second Lyapunov method; and the method of the Lyapunov vector function in stability theory and nonlinear analysis.
Article by: J J O'Connor and E F Robertson