李雅普诺夫Aleksandr Lyapunov

李雅普诺夫的母亲是Sofia Aleksandrovna Shilipova，父亲是Mikhail Vasilievich 李雅普诺夫。Mikhail Vasilievich是一位天文学家，在喀山大学工作，直到李雅普诺夫 Mikhailovich出生前两年，全家搬到了雅罗斯拉夫尔，因为他被任命为那里的Demidovski Lyceum的主任。Sofia Aleksandrovna和Mikhail Vasilievich的孩子们都很有才华，因为除了本传记的主人公外，他们还有两个男孩，其中一个（Sergei）成为作曲家，另一个（Boris）凭借其在斯拉夫语言方面的专长成为苏联科学院院士。

李雅普诺夫 Mikhailovich 先在家中接受教育，后来他的一个叔叔 R M Sechenov 为他进入文理中学做准备。李雅普诺夫并不是唯一一个由 Sechenov 辅导的人，Sechenov 同时也在教自己的女儿 Natalia Rafailovna Sechenov。事实上，Natalia 和李雅普诺夫多年后结婚，当时他29岁。李雅普诺夫的父亲去世几年后，Sofia Aleksandrovna 于1870年带着孩子们搬到下诺夫哥罗德（1932年至1990年称为高尔基），李雅普诺夫在该市进入文理中学。他于1876年毕业，进入圣彼得堡大学物理数学系，在那里他与马尔可夫成为朋友。

在圣彼得堡大学，他师从巴夫尼提·列波维奇·切比雪夫，正如我们将在下文看到的，巴夫尼提·列波维奇·切比雪夫对他产生了强烈影响。李雅普诺夫于1880年毕业，并留在圣彼得堡进行研究。他在1881年发表了两篇关于流体静力学的论文：On the equilibrium of heavy bodies in heavy liquids contained in a vessel of a certain shape和On the potential of hydrostatic pressures。次年，巴夫尼提·列波维奇·切比雪夫向李雅普诺夫提出了一个问题，这个问题将为他多年来的主要研究方向之一设定议程：-

已知在某一角速度下，椭球形式不再是旋转液体的平衡形式。在这种情况下，当角速度小幅增加时，它们难道不会转变为与椭球差别很小的某种新的平衡形式吗？

李雅普诺夫的硕士学位论文没有回答这个问题，但论文的工作正是由这个问题引发的。他于1884年提交了学位论文On the stability of ellipsoidal forms of equilibrium of a rotating liquid，并于次年圣彼得堡大学进行了答辩。此后，他被任命为哈尔科夫大学的privatdozent，在那里教授力学并继续为他的博士学位论文进行研究。他将博士学位论文The general problem of the stability of motion提交给莫斯科大学，并于1892年10月12日（按现代历法）答辩后获得博士学位。这篇论文的重要性在诸如[8]、[12]、[16]等多篇文章中得到强调，这些文章都是为了庆祝这一基础性贡献发表一百周年而写的。

次年，他被任命为哈尔科夫大学教授，直到1902年一直留在那里。在哈尔科夫大学期间，他在哈尔科夫数学会中发挥了重要作用，从1891年到1898年担任副主席，从1899年起担任主席，直到1902年离开哈尔科夫。他还编辑了Communications of the Kharkov Mathematical Society。

在[13]中，Pavlovskaya审视了李雅普诺夫关于我们上面引用的、由巴夫尼提·列波维奇·切比雪夫首次提出的问题的研究。由巴夫尼提·列波维奇·切比雪夫提出的关于在旋转角速度足够小的变化下，旋转流体除椭球形状外还存在平衡图形的问题，首先由李雅普诺夫在一阶近似下解决。他后来基于开尔文-彼得·格思里·泰特变分原理研究流体椭球的稳定性问题。他证明了稳定性的一个充分条件是势能的二阶及更高阶变分为正。李雅普诺夫承认，对一阶变分施加某些额外约束降低了他方法的普遍性，但写道：-

但在这方面，几乎没有任何其他研究方法可以说是完全令人满意的。

李雅普诺夫确立了随着旋转角速度的变化，科林·麦克劳林椭球转变为卡尔·古斯塔夫·雅各布·雅可比椭球。转变点是一个分岔椭球，在此情况下对应于一个卡尔·古斯塔夫·雅各布·雅可比旋转椭球。

1901年，李雅普诺夫当选为圣彼得堡的俄罗斯科学院，次年成为科学院应用数学院士。Grigorian写道[1]：-

李雅普诺夫全身心投入科学工作。他回到了巴夫尼提·列波维奇·切比雪夫留给他的问题，在一系列持续到他去世的广泛论文中，发展了旋转重液体平衡图形的理论。

1917年，李雅普诺夫离开圣彼得堡，到黑海沿岸的敖德萨大学任职。他在大学任教，但1918年春天，他妻子的健康开始迅速恶化。Natalia Rafailovna患有一种肺结核，李雅普诺夫看着她的健康衰退，非常不安。1918年10月31日，李雅普诺夫的妻子去世，当天晚些时候，李雅普诺夫开枪自杀。三天后，他在医院去世。

我们已经描述了李雅普诺夫的主要工作，即关于旋转液体理论的工作。然而，他的工作还有其他方面我们应该提及。其中之一当然是他在probability方面的贡献，他因为教授该学科的课程而对此产生兴趣。特别是在1900年和1901年发表的两篇论文中，他使用基于特征函数的技术证明了中心极限定理。另一个我们应该提及的贡献是，他担任了莱昂哈德·欧拉文集两卷的编辑。

他因杰出贡献而获得多项荣誉，当选为多个科学院成员，如Accademia dei Lincei（1909年）和法国科学院（1916年）。他还被授予圣彼得堡大学、哈尔科夫大学和喀山大学的荣誉成员。在他诞辰一百周年时，人们向他表达了各种敬意。例如，1957年6月6日，舍盖·索伯列夫在莫斯科向苏联科学院主席团、科学院技术科学部和物理科学部、莫斯科大学、Moscow Mathematical Society、科学院力学研究所和科学院自动学与遥控力学研究所的联席会议作了On the works of A M Lyapunov on potential theory讲座。该讲座的文本见[22]。

其他论文如[8]描述了李雅普诺夫对运动稳定性的贡献如何长期影响了该学科的发展。[8]中考虑的主题包括：稳定性，特别是临界点的稳定性；李雅普诺夫函数的构造和应用；泛函-微分方程的稳定性；第二李雅普诺夫方法；以及稳定性理论和非线性分析中李雅普诺夫向量函数方法。

Aleksandr Mikhailovich Lyapunov's mother was Sofia Aleksandrovna Shilipova and his father was Mikhail Vasilievich Lyapunov. Mikhail Vasilievich was an astronomer who worked at Kazan University until two years before Aleksandr Mikhailovich was born, when the family moved to Yaroslavl on his appointment as director of the Demidovski Lyceum there. Sofia Aleksandrovna and Mikhail Vasilievich had talented children for, in addition to the subject of this biography, they had two boys one of whom (Sergei) became a composer and the other (Boris) became a member of the Soviet Academy of Sciences through his expertise in Slavic languages.

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. He graduated in 1876 and entered the Faculty of Physics and Mathematics at St Petersburg University, where he became friends with Markov.

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 [8], [12], [16] 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 [13] 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 [1]:-

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 [22].

Other papers such as [8] 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 [8] 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.