His work in the
field of differential equations,
potential theory,
the stability of systems and
probability theory
is very important. His main preoccupations were
the stability of equilibria and the motion of
mechanical systems, the model theory for the
stability of uniform turbulent liquid, and
particles under the influence of
gravity.
His work in the field of
mathematical physics
was very important for subsequent advances of
this field. His work from
1898
About some questions, connected with
Dirichlet's
tasks (О некоторых вопросах, связанных с задачей
Дирихле) contains a study of the properties of
potential
around
charges
and
dipoles,
continuously distributed along any surface. His
work in this field is in close connection with
the work of Steklov. Lyapunov developed many
important approximative methods. His methods,
today named
Lyapunov methods,
which he developed in
1899,
make it possible to define the stability of sets
of ordinary differential equations. He
elaborated the modern rigorous theory of the
stability of a system, and the motion of a
mechanical system on the basis of a finite
number of parameters. In probability theory, he
generalised the works of Chebyshev and Markov,
and he finally proved the
Central limit theorem
using more common conditions than his
forerunners. The method he used for the proof is
today one of the foundations of probability
theory. From
1899
to
1902
he was a head of Kharkov mathematical society
and an editor of his News. On the
December 2,
1900
he was elected as a corresponding member of the
Russian Academy of Sciences, and on the
October 6,
1901
as a fully entitled member of the Academy in the
field of applied mathematics.
With his
researches on
celestial mechanics,
he opened a new page in the history of global
science, and showed the inaccuracy in the
knowledge of several foreign scientists. In
1908
he participated at the 4th Mathematical congress
in
Rome.
At this time he took part in the publication of
Euler's
selected works, and he was an editor of the 18th
and 19th part of this miscellany. By the end of
June
1917,
he went with his wife, who was seriously ill, to
his brother Boris in
Odessa,
Russia
(now
Ukraine).
His wife's impending death, his own partial
blindness, and the generally bad conditions for
life, all contributed to his anxiety. In spite
of this he delivered his last lecture about the
form of celestial bodies at the invitation of
the Department of Physics and Mathematics at
Odessa. On
October 31
his wife died, and on the same day he shot
himself. He then lay unconscious for a few days
till his death.
He usually
worked four to five hours at night, and many
times even the whole night. Once or twice he
visited the theatre, or went to some concert. He
had many students. But for the few who really
knew him, Lyapunov was a rather raptured man. He
had a lean figure, outwardly he acted pretty
rude, otherwise he had a hot-blooded and
sensitive temper. He was an honorary member of
many universities, an external member of the
Academy in Rome and a corresponding member of
the
Academy of Sciences
in
Paris.
His paper "problème
générale de la stabilité du mouvement" (1892)
(in French) marks the beginning of
stability theory.
(Resource : en.wikipedia.org)