阿列克谢·克雷洛夫Aleksey Krylov

阿列克谢·克雷洛夫的父亲是Nikolai Alexandrovich 克雷洛夫，一位退役炮兵军官。大约在克雷洛夫出生时，他的父亲是贵族副元帅，并成为阿拉特尔地区委员会的第一任主席。这个家庭肯定不富裕，但作为一位退伍军人的儿子，克雷洛夫有权接受免费教育。他于1878年进入圣彼得堡的海事高中。他于1884年毕业，获得优异成绩，并被分配到主要水文管理局的罗盘部门。在那里，他开始研究罗盘偏差，这是一个他会多次回到的主题。克雷洛夫在该部门的工作由Ivan Petrovich de Collong监督，他是一位杰出的科学家，也是罗盘磁偏差理论的创始人。De Collong于1875年发明了一种新型罗盘，并担任主要水文管理局局长。

1888年，克雷洛夫进入圣彼得堡海事学院的造船系。在那里，亚历山大·科尔金教他高等数学，亚历山大·科尔金是巴夫尼提·列波维奇·切比雪夫的学生，也是偏微分方程方面的专家。亚历山大·科尔金总是为那些他认定具有非凡能力的学生付出额外努力，给他们个别辅导，并提出非常困难且富有挑战性的问题。克雷洛夫从亚历山大·科尔金的辅导中受益匪浅，仅学习两年后便于1890年以一等成绩毕业。亚历山大·科尔金随后说服克雷洛夫留在海事学院，接替他教授课程。克雷洛夫照做了，留在那里任教近50年[1]：-

他在这所军事海事学院教授各种理论科学和工程科学约五十年，从学生中培养出一大批既是工程师又是科学家的造船师。

1898年，克雷洛夫因对船舶振荡运动理论的杰出贡献而获得皇家造船师学会颁发的金质奖章。

海军管理局拖曳水池建于1894年，用于进行船模试验，以确定达到规定航速所需的发动机功率，以及所需功率最小的船体线型。这是俄罗斯第一个用于测试船舶设计的实验水池，也是世界上第六个。水池的主管是A A Grekhnev，但他离开后，海军部上校A N 克雷洛夫于1900年1月3日被任命为'the Acting Superintendent of the Tank。克雷洛夫提议建立一个科学机构，其中包括拖曳水池、用于研究造船材料的测试与物理化学实验室、一个机械实验室和一个电气工程实验室。1902年4月3日，尼古拉二世皇帝参观了拖曳水池，并且：-

……对所做的实验和所观察到的事务秩序表示完全满意……

向作为水池主管的克雷洛夫表达了他的帝国谢意。克雷洛夫在拖曳水池的工作涵盖[1]：-

……浮力、稳性、横摇与纵摇、振动和性能的理论，以及罗盘理论。

这些正是莱昂哈德·欧拉所称的“海军科学”的主题。但克雷洛夫走得更远，于1902年登上一等巡洋舰Askold参加了一次航行。

1908年1月，此时已是将军的克雷洛夫被任命为首席海军舰船建造检查官和海事工程委员会主席[1]：-

他的勇气和正直导致他与海事部官员发生冲突，并拒绝为他们做进一步的工作。

在[6]中，Botchev给出了说明克雷洛夫与官员斗争的实例：-

为了推广他的创新，他常常不得不与高层官员的停滞和僵化观点作斗争。有一次，在一个重要会议上面对众多听众发表演讲时，克雷洛夫向海军军官们讲话，请求他们支持他“与造船业中的陈规作斗争”。当时作为一名海军军官，他因这次演讲而受到正式训斥。另一件事也相当显著地说明了克雷洛夫的个性：在一次高级技术委员会的会议上，克雷洛夫曾直接从船上带来几名技术人员，以便他们能在辩论中支持他的观点。

克雷洛夫于1910年辞去了海事部的职务。1914年，他被莫斯科大学授予应用数学荣誉博士学位。同年，他当选为俄罗斯科学院成员，并于1916年成为正式成员。1917年十月革命后，克雷洛夫站在苏维埃政府一边，但试图影响他们对科学的态度。1919年，他与其他一些人一起组织了俄罗斯物理学家协会。1919年2月，他们成立了一个由四人组成的委员会，成员包括Ioffe、克雷洛夫、Anri和Lazarev，以协调外国文献、仪器和设备的接收，并重启国外研究旅行。他们呼吁政府支持新研究所，重建俄罗斯与国外物理学家之间的联系，并恢复科学期刊的出版。事实上，克雷洛夫是革命后最早被允许前往西方的科学家之一，他于1921年前往伦敦，以重建俄罗斯与西方之间的联系。

从1927年到1932年，他是苏联科学院物理数学研究所的所长。1939年，他成为俄罗斯苏维埃联邦社会主义共和国的功勋科学家和工程师，并于1943年因其罗盘理论方面的工作而被授予国家奖，并被授予“社会主义劳动英雄”称号。

克雷洛夫在将数学应用于造船方面取得了许多数学进展。在流体动力学方面，在许多进展中，他对浅水中船舶运动理论做出了重大贡献。1904年，他构造了一个机械积分器来求解常微分方程，是俄罗斯第一个制造这种仪器的人。他在1905年的一篇论文中改进了约瑟夫·傅里叶求解边值问题的方法，并给出了许多应用[1]：-

在利用数学和力学发展他的船舶理论的同时，克雷洛夫也改进了这两个学科的方法。在一篇关于固定截面枢轴的受迫振动（1905年）的论文中，他提出了对约瑟夫·傅里叶求解边值问题方法的原创性发展，指出其适用于一系列重要问题：例如，蒸汽驱动机器指示器的理论、仪器管道中气体压力的测量，以及末端带有飞轮的滚子的扭转振动。

他在1912年的一篇论文中研究了Fourier series的收敛加速，并在1917年发表的一篇论文中研究微分方程的近似解。

1931年，他找到了一种求解确定机械系统振动频率的久期方程的新方法，该方法优于约瑟夫·拉格朗日、皮埃尔·西蒙·拉普拉斯、卡尔·古斯塔夫·雅各布·雅可比和于尔班·勒维耶的方法。这篇论文On the numerical solution of the equation by which, in technical matters, frequencies of small oscillations of material systems are determined涉及特征值问题。克雷洛夫在论文中写道（例如见[6]）：-

显然，如果对于k = 2和k = 3，容易组成这个[久期]方程，那么对于k = 4，展开就变得繁琐，而对于k大于5的值，直接方式完全无法实现。因此，应当使用避免行列式完全展开的方法。这篇论文的目的是……以展开形式呈现组成久期方程的简单方法，此后，其求解，即其根的数值计算，不会有任何困难。在我们描述这些方法之前，最好稍微回顾一下，考虑这些方法的最初创造者，约瑟夫·拉格朗日和皮埃尔·西蒙·拉普拉斯，以及后来像于尔班·勒维耶这样伟大的天文学家和像卡尔·古斯塔夫·雅各布·雅可比这样伟大的数学家是如何进行的……

这种对数学和力学历史的明显兴趣以多种方式体现在克雷洛夫的工作中[1]：-

克雷洛夫的实际兴趣与对十七、十八和十九世纪经典数学和力学思想与方法的深刻理解相结合；在艾萨克·牛顿、莱昂哈德·欧拉和卡尔·弗里德里希·高斯的世界中，他找到了适用于解决当代问题的被遗忘的方法。

事实上，克雷洛夫于1915年出版了艾萨克·牛顿的Philosophiae Naturalis Principia Mathematica的第一个从拉丁文翻译的俄译本。在克雷洛夫生前出版了三版，然后1989年出版了第三版的影印本。1933年10月5日，克雷洛夫在莫斯科苏联科学院举行的纪念莱昂哈德·欧拉逝世150周年的庆祝活动上发表了演讲。这篇演讲于1935年出版，演讲的保加利亚文译本于1983年出版，以纪念莱昂哈德·欧拉逝世200周年。

最后，让我们提及克雷洛夫出版的一些经典教材。Lectures on Approximate Calculations的第一版于1911年问世，第二版于1932年，第三版于1935年，第四版收入克雷洛夫的全集。On Some Differential Equations of Mathematical Physics Having Application to Technical Problems的第一版于1913年问世，第二版于1932年，第四版于1948年作为克雷洛夫全集的一部分出版。Vibration of ships于1936年首次出版，作为技术学校的造船学教科书。1943年，他出版了 Thoughts and materials on teaching mechanics. 。他在此书中提出的教学思想在[18]中有详细讨论。

克雷洛夫娶了Elisaveta Dmitrievna Dranitsyna；他们的女儿Anna后来嫁给了著名物理学家Pyotr Leonidovich Kapitsa，成为Anna Kapitsa，后者因在磁学和低温物理学方面的研究于1978年获得诺贝尔物理学奖。克雷洛夫在列宁格勒（现圣彼得堡）去世，葬于沃尔科沃公墓。

Aleksei Krylov's father was Nikolai Alexandrovich Krylov who was a retired artillery officer. Around the time that Aleksei was born, his father was Deputy Marshal of the Nobility and became the first president of the Alatyr District Council. The family were certainly not wealthy, but as the son of an army veteran, Aleksei was entitled to a free education. He entered the Maritime High School in St Petersburg in 1878. He graduated in 1884, awarded a distinction, and was appointed to the compass unit of the Main Hydrographic Administration. There he began work on compass deviation, a topic he would return to many times. Krylov's work in the unit was supervised by Ivan Petrovich de Collong who was an outstanding scientist and the founder of the theory of magnetic deviation of the compass. De Collong had invented a new type of compass in 1875 and was Head of the Main Hydrographic Administration.

In 1888 Krylov joined the department of ship construction of St Petersburg Maritime Academy. There he was taught advanced mathematics by Aleksandr Nikolaevich Korkin, a student of Chebyshev, who was an expert in partial differential equations. Korkin always put in extra effort for the students whom he recognised as having exceptional abilities, giving them personal tuition and posing them very difficult and challenging problems. Krylov benefited greatly from Korkin's tuition and he graduated First Class in 1890 after only two years of study. Korkin then persuaded Krylov to stay at the Maritime Academy and take over teaching his courses. This Krylov did, remaining there to teach for almost 50 years [1]:-

He taught various theoretical and engineering sciences for about fifty years at this military-maritime institute, creating from among his students a large school of shipbuilders who were both engineers and scientists.

In 1898 Krylov received a Gold Medal from the Royal Institution of Naval Architects for his outstanding contributions to the theory of oscillating motion of a ship.

The Naval Administration Towing Tank was set up in 1894 to perform ship model tests to find engine power requirements for specified speeds and hull lines which required the least power. It was the first experimental basin to test ship design in Russia and the sixth the world. The Superintendent of the Tank was A A Grekhnev but after he left, Captain of the Admiralty A N Krylov was appointed 'the Acting Superintendent of the Tank' on 3 January 1900. Krylov proposed establishing a scientific institution that would include the Towing Tank, testing and physical-chemical laboratories for research on ship construction materials, a mechanical laboratory and an electrical engineering laboratory. On 3 April 1902 Emperor Nicolas II visited the Towing Tank and:-

... deigned to be entirely pleased with both the performed experiments and the observed order of things ...

expressing his Imperial gratitude to Krylov as the Superintendent of the Tank. Krylov's work at the Towing Tank covered [1]:-

... theories of buoyancy, stability, rolling and pitching, vibration, and performance, and compass theories.

These were precisely the topics that Euler called 'naval science'. But Krylov went even further, joining a voyage on board the 1-rank cruiser Askold in 1902.

In January 1908, Krylov, by this time a General, was appointed Chief Naval Ship Construction Inspector and President of the Maritime Engineering Committee [1]:-

His courage and integrity led to conflicts with officials of the Maritime Ministry and his refusal to do further work for them.

in [6] Botchev gives examples which illustrate Krylov's struggle against officials:-

To promote his innovations, he often had to fight against stagnation and rigid views of the top officials. Once, giving a speech at an important meeting before a large audience, Krylov addressed naval officers asking them for their support in his "fight against the rut in shipbuilding". Being a naval officer at that time, he received an official reprimand for this speech. Another incident also quite remarkably illustrates Krylov's personality: on a sitting of a high-rank technical committee Krylov once took with him several technicians directly from the ships so that they could support his opinion in the debates.

Krylov gave up his posts at the Maritime Ministry in 1910. In 1914 he was awarded an honorary doctorate in applied mathematics from Moscow University. In the same year he was elected a member of the Russian Academy of Sciences, becoming a full member in 1916. After the October Revolution of 1917 Krylov sided with the Soviet Government but tried to influence their attitude towards science. In 1919, along with others, he organised the Russian Association of Physicists. In February 1919 they set up a committee of four, Ioffe, Krylov, Anri and Lazarev, to coordinate the receipt of foreign literature, instruments and equipment, and to restart foreign research travel. They called for government support for new institutes, the reestablishment of ties between physicists in Russia and abroad, and the resumption of publication of scientific journals. Indeed Krylov was one of the first scientists allowed to travel to the West after the Revolution, travelling to London in 1921 to re-establish contacts between Russia and the West.

From 1927 until 1932 he was director of the Physics-Mathematics Institute of the USSR Academy of Sciences. He became an honoured scientist and engineer of the Russian Soviet Federated Socialist Republic in 1939 and, in 1943, was awarded the State Prize for his work compass theory and made a "hero of socialist labour".

Krylov made many mathematical advances in his applications of mathematics to shipbuilding. In hydrodynamics, among many advances, he made significant contributions to the theory of ships moving in shallow water. In 1904 he constructed a mechanical integrator to solve ordinary differential equations, being the first in Russia to make such an instrument. He improved Fourier's method for solving boundary value problems in a 1905 paper and gave many applications [1]:-

While using mathematics and mechanics to work out his theory of ships, Krylov simultaneously improved the methods of both disciplines. In a paper on forced vibrations of fixed-section pivots (1905), he presented an original development of Fourier's method for solving boundary value problems, pointing out its applicability to a series of important questions: for example, the theory of steam-driven machine indicators, the measurement of gas pressure in the conduit of an instrument, and the twisting vibrations of a roller with a flywheel on its end.

He studied the acceleration of convergence of Fourier series in a paper in 1912, and studied the approximate solutions to differential equations in a paper published in 1917.

In 1931 he found a new method of solving the secular equation determining the frequency of vibrations in mechanical systems which is better than methods due to Lagrange, Laplace, Jacobi and Le Verrier. This paper On the numerical solution of the equation by which, in technical matters, frequencies of small oscillations of material systems are determined deals with eigenvalue problems. Krylov writes in the paper (see for example [6]):-

It is clear that, if for k = 2 and k = 3 it is easy to compose this [secular] equation, then for k = 4 the laying-out becomes cumbersome, and for values k more than 5 this is completely unrealisable in a direct way. Therefore one should use methods where the full development of the determinant is avoided. The aim of the paper ... is to present simple methods of composition of the secular equation in the developed form, after which, its solution, i.e. numerical computation of its roots, does not present any difficulty. Before we describe these methods, it is good to return a bit back and consider how the first creators of these methods, Lagrange and Laplace, and then such a great astronomer as Le Verrier and such a great mathematician as Jacobi proceeded ...

This obvious interest in the history of mathematics and mechanics come over in Krylov's work in a number of ways [1]:-

Krylov's practical interests were combined with a deep understanding of the ideas and methods of classical mathematics and mechanics of the seventeenth, eighteenth, and nineteenth centuries; and in the world of Newton, Euler, and Gauss, he found forgotten methods that were applicable to the solution of contemporary problems.

In fact Krylov published the first Russian translation from Latin of Newton's Philosophiae Naturalis Principia Mathematica in 1915. Three editions were published during Krylov's lifetime, then in 1989 a facsimile reproduction of the third edition was published. On 5 October 1933 Krylov gave a speech in Moscow during the celebration at the USSR Academy of Sciences of the 150th anniversary of the death of Euler. This speech was published in 1935, and a Bulgarian translation of the speech was published in 1983 on the occasion of the 200th anniversary of Euler's death.

Finally let us mention some of the classic texts which Krylov published. The first edition of Lectures on Approximate Calculations appeared in 1911, the second edition in 1932, the third in 1935, and the fourth in Krylov's collected works. The first edition of On Some Differential Equations of Mathematical Physics Having Application to Technical Problems appeared in 1913, the second edition in 1932, and the fourth appeared in 1948 as part of Krylov's collected works. Vibration of ships was first published in 1936 as a textbook in shipbuilding for Technical Schools. In 1943 he published Thoughts and materials on teaching mechanics. The ideas on teaching he presented in this text are considered in detail in [18].

Krylov married Elisaveta Dmitrievna Dranitsyna; their daughter Anna, later became Anna Kapitsa after marrying the famous physicist Pyotr Leonidovich Kapitsa who won the Nobel Prize for Physics in 1978 for his research in magnetism and low-temperature physics. Krylov died in Leningrad (now St Petersburg) and is buried in the Volkovo cemetery.