尤尔根·莫泽尔Jürgen Moser

尤尔根·莫泽尔的父母是Ilse Strehike和Kurt E 莫泽尔。纳粹上台时，莫泽尔只有五岁，但从那时起，他的父亲Kurt发现生活越来越艰难。Kurt Moser是一名神经科医生，并受到压力，要求他迎合纳粹的优生学观点。纳粹计划的一部分包括设立寄宿学校，用来培养纳粹德国未来的领导人。当莫泽尔十岁时，他的父母被鼓励把他送到这样一所学校，但Kurt充分了解情况，能够通过避免把莫泽尔送到那里，确保莫泽尔没有落入纳粹影响之下。莫泽尔在柯尼斯堡就读于文理中学，但当他十五岁时，所有男孩都被迫加入军事辅助部队，并接受操作高射炮的训练。莫泽尔身体不太强壮，所以被安排去做弹道计算工作。

1945年初，俄国军队向柯尼斯堡推进，并开始围困该城。莫泽尔和他班上其他十六岁的男孩一起被派去保卫城市，抵御俄国人。围困持续了两个月，在此期间，他班上除三人外全部阵亡。莫泽尔的哥哥也在这可怕的两个月里的战斗中丧生。莫泽尔每当有空闲时间就埋头于数学，以此逃避一些恐怖。1945年4月围困结束前，莫泽尔和他的父母乘船撤离，前往英国控制的区域。许多在围困中幸存下来的人现在却因船只被炮火击沉而丧生。莫泽尔和他的父母幸存下来，但在可怕的混乱中失散了。柯尼斯堡连同东普鲁士最北部的地区随后归苏联管辖，并于1946年更名为加里宁格勒。撤离六个月后，莫泽尔与他的父母取得了联系，此时他们已回到俄国人控制的区域。

莫泽尔试图越过边界进入俄国管区与父母团聚。然而，他在试图越境时被抓获并投入监狱。他被关押不久后便有了逃跑的机会，莫泽尔抓住了这个机会。随后他试图在俄国控制的区域继续学业，并寻求进入大学。然而当局不允许，于是莫泽尔再次冒险越境，这次是朝相反方向。他未能避免被发现，守卫边界的俄国军队开火，但幸运的是他未被弹雨击中，得以逃脱。现在回到西方控制的德国部分，他前往哥廷根，最终于1947年抵达该城。

哥廷根的弗朗兹·瑞利希一与莫泽尔交谈便立刻意识到，眼前是一个具有非凡数学天赋的人。莫泽尔没有钱的困难被克服了，他开始在导师弗朗兹·瑞利希的指导下研究微分方程谱理论。卡尔·西格尔曾是哥廷根大学的教授，但为了逃离他所憎恨的纳粹政权，于1940年自我流放去了美国。然而他于1951年回到哥廷根，并对莫泽尔产生了重大影响。特别是，莫泽尔通过卡尔·西格尔对天文学和数论产生了兴趣。1952年，莫泽尔凭借其学位论文Störungstheorie des kontinuierlichen Spektrums für gewöhnliche Differentialgleichungen zweiter OrdnungⓉ（二阶常微分方程连续谱的微扰理论）获得哥廷根大学博士学位。

莫泽尔随后发表了诸如Periodische Lösungen des restringierten Dreikörperproblems, die sich erst nach vielen Umläufen schliessenⓉ（限制性三体问题的周期解，多轮之后才接近）(1953)和Über periodische Lösungen kanonischer DifferentialgleichungssystemeⓉ（论微分方程正则系统的周期解）(1953)等论文。他留在哥廷根大学直到1953年，然后前往美国，凭借富布赖特奖学金在纽约大学度过了一年。他在理查·科朗特研究所度过了这一年，当时在那里工作的彼得·拉克斯写道：-

……我们意识到他非常特别，是男人中的王子，是身披闪亮铠甲的骑士。他具备所有德国美德：勤勉刻苦、热爱户外、热爱美、热爱音乐……与他一起做事是极好的陪伴，比如在山中徒步。……他热爱冒险，喜欢考验自己的能力；他极为自信。

在科朗特研究所工作一年后，他回到德国，在1954-55学年担任哥廷根卡尔·西格尔的助手。在此期间，他记录了卡尔·西格尔的讲座，这些笔记成为卡尔·西格尔1956年著作的基础，后来又成为Lectures on celestial mechanics合著书的基础，该书于1971年以德文出版，1995年出现英译本。1955年，莫泽尔的几篇论文发表，包括Singular perturbation of eigenvalue problems for linear differential equations of even order和Nonexistence of integrals for canonical systems of differential equations。此后莫泽尔移居美国。他在纽约经历了数学上富有启发的一年，但他还有另一个返回那里的理由，即理查·科朗特的女儿格特鲁德。他于1955年9月10日与Gertrude Courant结婚；他们有两个孩子尼娜和露西。莫泽尔于1955年被任命为纽约大学助理教授。然后他去了麻省理工学院，于1957年被任命为副教授。他于1960年回到麻省理工学院，当时被聘为正教授。他于1961年作为斯隆研究员度过，并从1967年到1970年担任科朗特研究所所长。科朗特研究所于1970年被抗议越南战争的示威者接管。他们试图摧毁计算机但失败了。这在多大程度上促使莫泽尔辞去所长职务尚不清楚，因为他从未非常喜欢这个角色。

1980年，莫泽尔离开美国，在苏黎世的Eidgenössische Technische Hochschule担任了一个职位。这次任命的方式相当奇怪，因为最初莫泽尔为该校一个空缺职位的申请人写了一封支持信。ETH的校长询问了推荐信的撰写者，并被告知他是世界上研究动力系统的领军人物。校长随后决定尝试任命莫泽尔到ETH。当然，人们可能会问，在如此成功的职业生涯之后，他为什么会离开纽约。几乎可以肯定，这是他在不真正回到德国的情况下所能做到的最接近回家的选择，因为战争的记忆仍然令人伤痛。从1984年到1995年退休，他一直担任ETH数学研究所的所长。

莫泽尔研究常微分方程、偏微分方程、谱理论、天体力学和稳定性理论。关于他对动力学的贡献，[5]的作者写道：——

他总是对他人的工作怀有浓厚兴趣，能够辨别根本趋势，并且总是做出必不可少的、往往是根本性的贡献。……

我们无法想象在1960年之后的时期里，还有另一位数学家对动力学中几乎所有主要趋势拥有如此广阔的视野和全面的理解，并对其发展产生了类似程度的影响。...

在他的工作中，他通常寻求的是智慧而非简单的知识，因此他强烈强调方法和洞见的发展，而不是将某个特定结果推向极限。...

莫泽尔几乎所有动力学工作的主导主题，是寻找动力系统中关于初始条件或系统扰动的稳定行为要素。

他对这一领域的主要贡献之一是在1962年，当时他在Nachrichten der Akademie der Wissenschaften Göttingen上发表了On invariant curves of area-preserving mappings of an annulus。这项工作引入了几乎可应用于任何哈密顿型动力系统的技术，以及“莫泽尔扭转稳定性定理”。当与弗拉基米尔·阿诺尔德的工作结合时，这导致了今天所谓的KAM理论。基于安德雷·柯尔莫哥洛夫的初始思想——这些思想在他1954年向国际大会发表的著名演讲中提出——这一理论为天体力学中的稳定性问题提供了一种惊人的新方法。

莫泽尔写了几本重要的书。首先我们提到Lectures on Hamiltonian systems（1968），它考察了临界点附近哈密顿系统的解的稳定性、幂级数展开的收敛性以及积分问题。接下来是Stable and random motions in dynamical systems（1973，2001年重印），它描述了在解析保守微分方程系统中，稳定行为和统计行为如何同时发生。Integrable Hamiltonian systems and spectral theory（1983）源自莫泽尔1981年在比萨高等师范学校所授的一门课程。在这里，莫泽尔考察了一维埃尔温·薛定谔方程的逆谱理论，正如他在引言中所写，其目的是表明：-

... 找到所有以有限多个区间为其谱的几乎周期势，等价于研究椭球面上的测地线。

莫泽尔1988年春季在苏黎世联邦理工学院所授的一门课程成为Selected chapters in the calculus of variations（2003）的基础。1979-80年，莫泽尔和Eduard Zehnder开始撰写一本关于哈密顿动力系统的书。他们的目标是写一本带有完整证明的入门教材，用物理学和天体力学中的例子来说明理论。他们从未完成这本书，只写了计划五章中的前三章。这三章于2005年出版。缺失的最后两章本应关于KAM理论和不稳定双曲解。

莫泽尔因其杰出贡献获得了许多荣誉。American Mathematical Society于1968年授予他乔治·戴维·伯克霍夫应用数学奖：-

... 表彰他对哈密顿动力系统理论的贡献，特别是他关于具有两个自由度的哈密顿系统周期解稳定性的证明，以及他与此项工作相关的思想的具体应用。

他于1973年当选为国家科学院会士，并因其对动力天文学的贡献于1969年被授予该机构的Craig Watson奖章。他于1983年至1986年担任国际数学联盟主席。同年，他被格罗宁根荷兰科学学会授予勒伊岑·布劳威尔奖章。他于1992年被德国数学会授予格奥尔格·康托尔奖章，并于1994年获得沃尔夫数学奖：-

因其在哈密顿力学稳定性方面的基础性工作，以及他对非线性微分方程深刻而有影响的贡献。

1996年，他成为伦敦数学会的名誉成员。莫泽尔于1973年应邀在达拉斯作美国数学会的约西亚·威拉德·吉布斯讲座，1975年在ETH作沃尔夫冈·泡利讲座，1976年在多伦多作美国数学会讨论会讲座，1977年在剑桥作戈弗雷·哈罗德·哈代讲座，1981年在比萨作恩里科·费米讲座，1984年在西雅图作工业与应用数学学会的冯·诺伊曼讲座。

1998年，莫泽尔应邀在柏林国际数学家大会上作报告，这是他第三次应邀在国际数学家大会上作报告。他作了一场Dynamical systems - past and present讲座，其中包含对KAM理论的历史回顾，随后是其对粒子加速器的应用。讲座讨论了关于地球磁场中带电粒子的卡尔·斯特默问题，也讨论了乔治·威廉·希尔的月球理论。可积系统也得到了讨论。

Paul Rabinowitz是莫泽尔在纽约的博士生之一，他在[7]中写道：-

对于认识他的人来说，莫泽尔体现了一位有创造力的科学家，或许更重要的是，一个真正的人。他的标准很高，品味无可挑剔。他的论文写得优雅。他不仅专注于自己的研究，还以多种方式为数学的福祉成功工作。他通过对年轻一代问题的深刻洞察——无论是科学问题还是其他问题——以及他温暖而明智的建议、关心和鼓励，激励了好几代年轻人。

莫泽尔在柏林发表演讲后仅活了一年多。他因前列腺癌去世，享年71岁。

Jürgen Moser's parents were Ilse Strehike and Kurt E Moser. Jürgen was only five years old when the Nazis came to power but from that time on his father Kurt found life increasingly difficult. Kurt Moser was a neurologist and came under pressure to fit in with the Nazi views on eugenics. Part of the Nazi programme involved setting up boarding schools where future leaders of Nazi Germany would be trained. When Jürgen was ten years old his parents were encouraged to send him to such a school but Kurt, fully aware of the situation, was able to ensure that Jürgen did not fall under the Nazi influence by avoiding having Jürgen sent there. Jürgen attended the Gymnasium in Königsberg but when he was fifteen years old all the boys were forced into a military auxiliary force and trained to man anti-aircraft guns. Jürgen was not very strong physically so he was put to work computing trajectories.

In early 1945 the Russian Army advanced towards Königsberg and began a siege of the city. Moser, along with the other sixteen year old boys from his class, were sent to defend the city from the Russians. The siege lasted for two months during which time all but three from his class were killed. Moser's elder brother also died in the fighting during these terrible two months. Moser escaped from some of the horrors by occupying himself in mathematics whenever he had a free moment. Before the siege ended in April 1945 Moser and his parents were evacuated by boat towards the British held zone. Many of those who had survived the siege now perished as boats were sunk by gunfire. Moser and his parents survived but were separated in the terrible chaos. Along with the extreme northern sector of East Prussia, Königsberg then passed to the sovereignty of the U.S.S.R. and was renamed Kaliningrad in 1946. Six months after the evacuation Moser made contact with his parents who, by this time, were back in the zone held by the Russians.

Moser tried to cross the border into the Russian sector to reunite with his parents. However he was caught as he attempted to cross the border and put in prison. He had not been held for long when there was an opportunity to escape which Moser took. He then attempted to continue his education in the Russian controlled zone and sought to enter university. The authorities did not allow this however so Moser again took the risk of crossing the border again, this time in the opposite direction. He did not manage to avoid being seen and the Russian troops guarding the border opened fire but by good fortune he was not hit by the hail of bullets and escaped. Now back in the part of Germany controlled by the West, he made his way to Göttingen eventually reaching the city in 1947.

Franz Rellich at Göttingen knew immediately that he began to talk to Moser that here was someone with an exceptional mathematical talent. The difficulty that Moser had no money was overcome and he began to study the spectral theory of differential equations with Rellich as his advisor. Carl Siegel had been a professor at Göttingen but had gone to the United States in self-imposed exile in 1940 to escape the Nazi regime which he hated. He returned to Göttingen, however, in 1951 and became a major influence on Moser. In particular Moser acquired an interest in astronomy and number theory through Siegel. In 1952 Moser was awarded his doctorate from Göttingen for his thesis Störungstheorie des kontinuierlichen Spektrums für gewöhnliche Differentialgleichungen zweiter Ordnung Ⓣ.

Moser then published papers such as Periodische Lösungen des restringierten Dreikörperproblems, die sich erst nach vielen Umläufen schliessen Ⓣ (1953) and Über periodische Lösungen kanonischer Differentialgleichungssysteme Ⓣ (1953). He remained at Göttingen University until 1953 when he went to the United States spending a year at New York University on a Fulbright Scholarship. He spent the year at the Courant Institute and Peter Lax, who was working there at the time, wrote this:-

... we realised that he was very special, a prince among men, a knight in shining armour. He had all the German virtues: devotion to hard work, a love of the outdoors, a love of beauty, of music ... He was exceedingly good company to do things with, like hiking in the mountains. ... He loved adventure and to test his powers; he had great self-confidence.

After working for a year at the Courant Institute he returned to Germany where he was an assistant to Carl Siegel at Göttingen in the academic year 1954-55. During this period he took notes of Siegel's lectures which became the basis for Siegel's 1956 book, then later became the basis of a joint book Lectures on celestial mechanics published in German in 1971 with an English translation appearing in 1995. In 1955 several of Moser's papers were published including Singular perturbation of eigenvalue problems for linear differential equations of even order, and Nonexistence of integrals for canonical systems of differential equations. After this Moser emigrated to the United States. He had experienced a mathematically stimulating year in New York, but he had another reason to return there, namely Richard Courant's daughter Gertrude. He married Gertrude Courant on 10 September 1955; they had two children Nina and Lucy. Moser was appointed to New York University as an assistant professor in 1955. Then he went to the Massachusetts Institute of Technology where he was appointed as an associate professor in 1957. He returned to the Massachusetts Institute of Technology in 1960 when he was made a full professor. He spent 1961 as a Sloan Fellow and, from 1967 to 1970, he served as Director of the Courant Institute. The Courant Institute was taken over by demonstrators protesting against the Vietnam war in 1970. They tried to destroy the computer but failed. How much this prompted Moser to step down as director is not clear for he had never greatly liked the role.

In 1980 Moser left the United States and took up a position at the Eidgenössische Technische Hochschule in Zürich. The appointment came about in a rather strange way, for initially Moser had written a letter of support for an applicant for a vacant position there. The president of ETH asked about the writer of the reference and was told he was the leading person in the world working on dynamical systems. The president then decided that he would try to appoint Moser to ETH. Of course one might ask why he would leave New York after such a successful career. Almost certainly it was as close as he could come to returning home without actually going back to Germany where memories of the war still hurt. From 1984 until his retirement in 1995 he was the Director of the Mathematics Research Institute at the ETH.

Moser worked in ordinary differential equations, partial differential equations, spectral theory, celestial mechanics, and stability theory. Writing about his contributions to dynamics, the authors of [5] write:-

Always keenly interested in the work of others, he was able to discern the fundamental trends and invariably made essential, often fundamental, contributions. ...

We cannot think of another mathematician in the period after 1960 who had such a broad view and comprehensive understanding of virtually all major trends in dynamics and influenced their development to a similar degree. ...

In his work he usually searched for wisdom rather than simply knowledge, and thus he strongly emphasized developments of methods and insights over pushing a specific result to the limit. ...

The leading theme of virtually all of Moser's work in dynamics is the search for elements of stable behaviour in dynamical systems with respect to either initial conditions or perturbations of the system.

One of his major contributions to this area was in 1962 when he published On invariant curves of area-preserving mappings of an annulus in the Nachrichten der Akademie der Wissenschaften Göttingen. This work introduced techniques which could be applied to almost any dynamical system of Hamiltonian type and the "Moser twist stability theorem". When combined with the work of Arnold this led to what is today called KAM Theory. Based on initial ideas by Kolmogorov, presented in his famous address to the International Congress in 1954, this theory provided a stunning new approach to stability problems in celestial mechanics.

Moser wrote several important books. First we mention Lectures on Hamiltonian systems (1968) which examines problems of the stability of solutions, the convergence of power series expansions, and integrals for Hamiltonian systems near a critical point. Next is Stable and random motions in dynamical systems (1973, reprinted 2001) which describes how stable behaviour and statistical behaviour take place together in analytic conservative systems of differential equations. Integrable Hamiltonian systems and spectral theory (1983) arises from a course of lectures which Moser gave at the Scuola Normale Superiore in Pisa in 1981. Here Moser examines inverse spectral theory for the one-dimensional Schrödinger equation with the aim, as he writes in the introduction, of showing that:-

... finding all almost periodic potentials having finitely many intervals as its spectrum is equivalent to the study of the geodesics on the ellipsoid.

A course of lectures that Moser gave at ETH in the spring of 1988 became the basis for Selected chapters in the calculus of variations (2003). In 1979-80 Jürgen Moser and Eduard Zehnder began to write a book on Hamiltonian dynamical systems. They aimed to write an introductory text with complete proofs using examples from physics and celestial mechanics to illustrate the theory. They never finished the book, only writing the first three of five planned chapters. These three chapters were published in 2005. The missing final two chapters would have been on KAM theory and unstable hyperbolic solutions.

Moser received a host of honours for his remarkable contributions. The American Mathematical Society awarded him their D George Birkhoff Prize in Applied Mathematics in 1968:-

... for his contributions to the theory of Hamiltonian dynamical systems, especially his proof of the stability of periodic solutions of Hamiltonian systems having two degrees of freedom and his specific applications of the ideas in connection with this work.

Elected to the National Academy of Sciences in 1973, he had been awarded its Craig Watson Medal in 1969 for his contributions to dynamic astronomy. He served as president of the International Mathematical Union from 1983 to 1986. In the same year he was awarded the L E J Brouwer Medal by the Dutch Scientific Society of Groningen. He was awarded the Georg Cantor Medal by the German Mathematical Society in 1992, and received the Wolf Prize in Mathematics in 1994:-

For his fundamental work on stability in Hamiltonian mechanics and his profound and influential contributions to nonlinear differential equations.

In 1996 he was made an honorary members of the London Mathematical Society. Moser was invited to give the Gibbs lecture of the American Mathematical Society in Dallas in 1973, the Pauli lectures at ETH in 1975, the American Mathematical Society Colloquium lectures in Toronto in 1976, the Hardy lectures in Cambridge in 1977, the Fermi lectures in Pisa in 1981, and the John von Neumann Lecture of the Society for Industrial and Applied Mathematics in Seattle in 1984.

In 1998 Moser was invited to address the International Congress of Mathematicians in Berlin, the third time he had been invited to address an International Congress. He delivered a lecture Dynamical systems - past and present which contains an historical review of KAM theory followed by applications to particle accelerators. The Störmer problem concerning charged particles in the Earth's magnetic field is discussed as is Hill's lunar theory. Integrable systems are also discussed.

Paul Rabinowitz, one of Moser's doctoral students in New York, writes in [7]:-

To those who knew him Moser exemplified a creative scientist and, perhaps even more important, a human being. His standards were high and his taste impeccable. His papers were elegantly written. Not merely focused on his own research, he worked successfully for the well-being of mathematics in many ways. He stimulated several generations of younger people by his penetrating insights into their problems, scientific and otherwise, and by his warm and wise counsel, concern, and encouragement.

Sadly Moser had only just over a year to live after delivering his lecture in Berlin. He died of prostate cancer at the age of 71.