# Jürgen Kurt Moser

### Quick Info

Königsberg, Germany (now Kaliningrad, Russia)

Zürich, Switzerland

**Jürgen Moser**was a German-born American mathematician who worked in Hamiltonian dynamical systems and partial differential equations.

### Biography

**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. ...One of his major contributions to this area was in 1962 when he published

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.

*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.

### References (show)

- V Arnold, Jürgen Moser (1928-1999). Déclin des mathématiques (après la mort de Jürgen Moser),
*Gaz. Math.*No.**84**(2000), 92-95. - Dedicated to the memory of Jürgen K Moser, Comm. Pure Appl. Math.
**56**(7) (2003), i-ii; 813--1028. - Dedicated to the memory of Jürgen K Moser, Comm. Pure Appl. Math.
**56**(8) (2003), i-ii; 1029-1245. - A Giorgilli, A tribute to Jürgen Moser (1928-1999),
*Celestial Mech. Dynam. Astronom.***77**(3) (2000), 153-155. - B Hasselblatt and A Katok, The development of dynamics in the 20th century and the contribution of Jürgen Moser,
*Ergodic Theory Dynam. Systems***22**(5) (2002), 1343-1364. - P D Lax, Jürgen Moser, 1928-1999,
*Ergodic Theory Dynam. Systems***22**(5) (2002), 1337-1342. - J N Mather, H P McKean, L Nirenberg and P H Rabinowitz, Jürgen K Moser (1928-1999),
*Notices Amer. Math. Soc.***47**(11) (2000), 1392-1405. - J Moser, Recollections. Concerning the early development of KAM theory, in
*The Arnoldfest, Toronto, ON, 1997*(Amer. Math. Soc., Providence, RI, 1999), 19-21. - J Moser to the 70th anniversary : Curriculum vitae : J Moser at 70 (Russian),
*Regul. Chaotic Dyn.***3**(3) (1998), 3-9. - E J Zehnder, Hommage à Jürgen Moser,
*Gaz. Math.*No.**59**(1994), 57-66. - E J Zehnder, Cantor-Medaille für Jürgen Moser,
*Jahresber. Deutsch. Math.-Verein.***95**(2) (1993), 85-94.

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Written by J J O'Connor and E F Robertson

Last Update November 2006

Last Update November 2006