# Aleksandr Mikhailovich Liapunov

## Liapunov, Aleksandr Mikhailovich

Born May 25 (June 6), 1857, in laroslavP; died Nov. 3, 1918, in Odessa. Russian mathematician and engineer. Academician of the St. Petersburg Academy of Sciences (1901; corresponding member, 1900). Student of P. L. Chebyshev.

Liapunov graduated from the University of St. Petersburg in 1880. From 1885 he was a docent and from 1892 a professor at the University of Kharkov. In 1902 he began working at the St. Petersburg Academy of Sciences.

Liapunov created the modern rigorous theory of the stability of the equilibrium and motion of mechanical systems defined by a finite number of parameters. From the mathematical aspect, this problem reduces to the investigation of the limiting behavior of solutions of systems of ordinary differential equations as the independent variable approaches infinity. Liapunov defined stability with respect to disturbances of the initial data of the motion. Before Liapunov, problems of stability were usually solved to a first approximation, that is, by rejecting all nonlinear terms of the equations of motion without explaining the validity of such a linearization of the equations.

An outstanding achievement of Liapunov was the setting up of a general method for solving stability problems. His chief work was his doctoral dissertation, *General Problem of the Stability of Motion* (1892). In this work he rigorously defined the fundamental concepts of the theory of stability, indicated cases when an examination of the linear equations of the first approximation yields a solution of the stability problem, and studied in detail certain important cases that occur when the first approximation does not provide an answer to this question. Liapunov’s dissertation and succeeding works in the field contain a whole series of fundamental results in the theory of both linear and nonlinear ordinary differential equations.

Liapunov devoted numerous studies to the theory of the equilibrium figures of a uniformly rotating liquid whose particles attract each other according to the law of universal gravitation. Before Liapunov, ellipsoidal equilibrium figures had been established for a homogeneous liquid. Liapunov was the first to prove the existence of equilibrium figures close to ellipsoidal for a homogeneous and a weakly nonhomogeneous liquid. He established that nonellipsoidal equilibrium figures of a homogeneous liquid are derived from some ellipsoidal equilibrium figures close to them and that equilibrium figures of a weakly nonhomogeneous liquid are derived from other ellipsoidal equilibrium figures. He also solved a problem proposed to him at the beginning of his career by P. L. Chebyshev concerning the possibility of deriving nonellipsoidal equilibrium figures from an ellipsoidal equilibrium figure possessing the greatest (possible for ellipsoids) angular velocity. The answer proved to be negative.

Liapunov was the first to prove rigorously the existence of equilibrium figures of a slowly rotating nonhomogeneous liquid that are close to a sphere under extremely general assumptions regarding the variation of density with depth. Liapunov also undertook a study of the stability of both ellipsoidal figures and new figures he discovered for the case of a homogeneous liquid. The very formulation of the stability problem for a continuous medium (liquid) was unclear before Liapunov. He was the first to rigorously formulate the problem and by means of a careful mathematical analysis investigated the stability of equilibrium figures. In particular, he proved the instability of so-called pear-shaped equilibrium figures and thus disproved the contrary assertion of the British astronomer G. Darwin. Liapunov’s works on the equilibrium figures of a rotating liquid and on the stability of these figures occupy a central place in the theory of equilibrium figures as a whole.

Though small in number, Liapunov’s works on some problems of mathematical physics were extremely important for the further development of science. Of fundamental importance is his study *Some Questions Connected With the Dirichlet Problem* (1898). This work was based on a study of the properties of the potential arising from charges and dipoles continuously distributed on a given surface. The study of the so-called double layer potential (the case of dipoles) is of the greatest importance. Liapunov later obtained important results related to the behavior of the derivatives of the solution of the Dirichlet problem in approaching a surface for which the boundary condition is given. Proceeding along these lines, he was the first to demonstrate the symmetry of Green’s function for the Dirichlet problem and to prove the formula giving the solution of the problem in the form of a surface integral of the product of the function occurring in the boundary condition and the normal derivative of Green’s function. Liapunov imposed certain constraints on the boundary surface under all these conditions. Surfaces that satisfy them are now called Liapunov surfaces.

In probability theory Liapunov proposed a new method of study (the method of “characteristic functions”) that is remarkable in its generality and productiveness. Generalizing the studies of P. L. Chebyshev and A. A. Markov (the elder), Liapunov proved the central limit theorem of probability theory under much more general conditions than his predecessors.

### WORKS

*Obshchaia zadacha ob ustoichivosti dvizheniia*. Moscow-Leningrad, 1950.

*Izbr. trudy*. Edited by V. I. Smirnov. Leningrad, 1948. (Contains a bibliography and literature about Liapunov.)

*Sobr. soch*., vols. 1-5. Moscow, 1954-65.

### REFERENCES

*Materialy dlia biograficheskogo slovaria deistvitel’nykh chlenov Akademii Nauk*, part I. St. Petersburg, 1915. (

*Imp. Akademiia Nauk 1889-1914*, vol. 3.)

Liapunov, B. M.

*Kratkii ocherk zhizni i deiatel’nosti A. M. Liapunova*. Leningrad, 1930.

*Aleksandr Mikhailovich Liapunov*. A bibliography compiled by A. M. Lukomskaia. Moscow-Leningrad, 1953.