# Moser, Jürgen K.

# MOSER, JüRGEN K.

(*b*. Königsberg, Germany (later Russia), 4 July 1928; *d*. Zürich, Switzerland, 17 December 1999)

*mathematics, analysis, celestial mechanics*.

Moser won a Wolf Prize, awarded by the Wolf Foundation in 1994–95, “for his fundamental work on stability in Hamiltonian mechanics and his profound and influential contributions to nonlinear differential equations.” He was one of the founders of KAM (Kol-mogorov-Arnold-Moser) theory and of Nash-Moser theory. His book, *Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics* (1973), helped redefine celestial mechanics.

A profound mathematical analyst, Moser did deep work in a variety of fields of mathematics, both pure and applied. These included dynamical systems, especially small divisor problems and relations with celestial mechanics; functional analysis, especially Nash-Moser theory; partial differential equations, especially regularity questions and Harnack inequalities; complex geometry; completely integrable systems; and variational calculus.

**Early Years.** Moser’s father was a neurologist. His parents managed to prevent his being enrolled in an “elite” school for future Nazi leaders at the age of ten, but he was drafted to fight on the eastern front at the age of fifteen. After surviving the war and escaping from the Russian zone of postwar Germany, he enrolled in Göttingen in 1947, where he studied mathematics under Franz Rellich.

Later, Moser worked with Carl Ludwig Siegel, one of whose interests was celestial mechanics. Siegel’s book, *Vorlesungen über Himmelsmechanik*, was based on notes that Moser took of Siegel’s lectures. A revision of this book by Moser, *Lectures in Celestial Mechanics*, appeared under the joint authorship of Siegel and Moser. Siegel suggested that Moser work on Birkhoff’s problem related to the stability of the solar system.

**Stability of the Solar System.** Have the orbits of the planets remained approximately the same since the formation of the solar system from four billion to five billion years ago? Isaac Newton’s theory predicts that a planet, in the absence of other planets, would move on an elliptical orbit around the sun. It also predicts that these elliptical orbits change slowly over time under the gravitational influence of the other planets. It was observed in the eighteenth century that perturbations of the orbits of the planets were large enough that the solar system might become unstable in a relatively short time, compared to the age of the solar system. Toward the end of the eighteenth century, Joseph-Louis Lagrange and Pierre-Simon Laplace showed that all the observed perturbations could be explained by first and second order perturbation theory on the basis of Newton’s law of gravitation. Such perturbations were quasi-periodic. This showed that the observed perturbations were consistent with the stability of the solar system.

First and second order perturbation theory provided only the beginning terms of an infinite series of periodicities. Karl Weierstrass observed that if this series converged, then the solar system (or, rather, Newton’s model of it) would be stable. He suggested to Gösta MittagLeffler that the problem of proving convergence should be sponsored as a prize by the king of Sweden. Henri Poincaré won the prize for his paper “Sur le problème des trois corps et les équations de la dynamique.”

Poincaré did not actually solve the problem; rather, the prize was awarded for the paper’s wealth of ideas. He did, however, find an argument that strongly suggested the existence of random motions (“chaos” in the sense of chaos theory). Such random motions are incompatible with the convergence of all the series that arise, although they are compatible with some, even most.

**KAM Theory.** In 1954 Andrey Kolmogorov published “On the Conservation of Conditionally Periodic Motions for a Small Change in Hamilton’s Function.” This was the first progress toward proving stability. His theorem states that in a small Hamiltonian perturbation of an integrable system satisfying a nondegeneracy condition, most invariant tori survive. An integrable system is one in which all invariant solutions are quasi-periodic, that is, can be expressed as convergent infinite sums of periodic functions. Every quasi-periodic solution lies on an invariant torus and passes arbitrarily close to every point of the torus infinitely often.

Saying that an invariant torus survives means that in the perturbed system, there is an invariant torus near the original invariant torus, and every solution in it is quasi-periodic with the same frequencies as the solutions in the unperturbed torus. Saying that most invariant tori survive means that the invariant tori in the perturbed system fill out most of the phase space—that is, their complement has small measure (2*d*–dimensional volume, where *d* is the number of degrees of freedom).

Quasi-periodic motions are very regular. Thus, the union of the invariant tori exhibits no chaos. On the other hand, the complement of this set generally does exhibit chaos, as Poincaré’s argument and later extensions of it show.

Planetary motion may be regarded as a small perturbation of an integrable system, provided that the masses of the planets are small enough in relation to the mass of the sun. The integrable system in this case is the limiting case when the masses of the planets vanish. Unfortunately, Kolmogorov’s theorem does not apply, because his nondegeneracy condition is not satisfied. Nonetheless, in the

early 1960s there were high hopes of extending Kolmogorov’s method to the problem of stability of planetary motions. In fact, Vladimir I. Arnold, Moser, and much later, Michael Herman and Jacques Féjoz, obtained results along these lines, thus realizing these hopes, although the extension was much harder than anticipated.

In his prize paper, Poincaré studied the restricted three-body problem extensively. This is the case where one mass vanishes (so it moves under the influence of the other two masses but does not influence them) and the other two masses move in circular orbits about each other. It is further assumed that all three masses move in a plane. Poincaré showed that the study of such a system can be largely reduced to the study of the dynamics of an area preserving transformation of an annular region in the plane. This led Poincaré and, later, George D. Birkhoff in the early twentieth century, to study the dynamics of area-preserving transformations in great depth.

In his paper, “On Invariant Curves of Area-Preserving Mappings of an Annulus” (1962), Moser solved the fundamental problem posed by Birkhoff in his study of area-preserving transformations of surfaces. This concerned the stability of a fixed point *P*. Moser’s theorem states that there exist arbitrarily small invariant neighborhoods of *P*, under suitable hypotheses on the transformation. His proof was inspired by the work of Kolmogorov but also relied on a normal form theorem of Birkhoff and a very profound analytic technique originally developed by John Nash to solve the isometric embedding problem. Moser followed up this paper with three papers in 1966 and 1968, one of them written jointly with William H. Jefferys, regarding the restricted three-body problem and a problem concerning confinement in a magnetic bottle. Together with a deep result of Arnold (the stability of a planetary system in the plane with only two planets, both of extremely small mass), these were the first stability results in celestial mechanics.

Although all of these results closely resemble Kolmogorov’s original result, it does not seem that Kolmogorov’s original method can be used to obtain them without the profound improvements introduced by Moser and Arnold. Moser’s improvement used Nash’s method, which enabled him to deal with the case when the given data are highly differentiable, instead of analytic, as in the Kolmogorov case. This is usually taken as the main significance of Moser’s improvement. Nonetheless, for the solution of Birkhoff’s problem, Moser’s improvement seems necessary even when the transformation is analytic. This is because Moser’s solution depends on Birkhoff normal form, which presents the given transformation as a small perturbation of an integrable system. The perturbation is small in the *C*^{4} topology, which is good enough for the application of Moser’s method, but not for the application of Kolmogorov’s method, which would require that the perturbation be small in the *C ^{ω}* topology. Likewise, the extra robustness of Moser’s method seems to be required for the problems in celestial mechanics and the problem of magnetic confinement that he solved.

**Nash-Moser Theory.** Both Kolmogorov’s stability theorem and Nash’s isometric embedding theorem depend on deep extensions of the classical implicit function theorem. Kolmogorov’s proof used Newton’s iteration scheme. Nash’s involved smoothing of functions. In his solution of Birkhoff’s problem and in “A New Technique for the Construction of Solutions of Nonlinear Differential Equations” (1961) Moser combined these two techniques.

In lectures at Pisa in 1966, Moser showed that Newton iteration with smoothing can be used to prove infinite dimensional implicit function theorems in many circumstances where classical methods fail. These ideas were later developed into abstract implicit function theorems by Francis Sergeraert and Richard S. Hamilton. Such abstract implicit function theorems are known as Nash-Moser theory. The paper of Féjoz, “Démonstration du ‘théorème d’Arnold’ sur la stabilité du système planétaire (d’après Herman),” which generalized Arnold’s theorem on planetary stability to an arbitrary number of planets in arbitrary dimensions, relies heavily on such abstract implicit function theorems.

**More Analysis.** In addition to his contributions to KAM theory and Nash-Moser theory, Moser made major contributions to other fields of mathematical analysis. In 1961 Moser generalized the classical Harnack inequality in the theory of linear elliptic partial differential equations. Moser’s result assumed less regularity on the coefficients than the classical result. A key estimate obtained independently by Ennio De Giorgi and Nash in their solutions of Hilbert’s nineteenth problem follows easily from Moser’s result. De Giorgi and Nash had given elaborate (and different) proofs of this estimate about three years earlier. Moser also deduced a generalization of Bernstein’s theorem (concerning minimal surfaces in three-dimensional space) to arbitrary dimensions.

In 1971 Moser proved a sharp form of an inequality due to Neil Trudinger and used it to determine which even functions on the two spheres are the Gauss curvature of a metric conformal to the standard metric. In 1974, Shiing-Shen Chern and Moser determined local invariants of real hypersurfaces of real codimension one in a complex number space of arbitrary dimension. In the 1980s Moser created a theory of foliations and laminations of codimension one on a torus of arbitrary dimension whose leaves are minimals of a nonlinear variational problem (for example, minimizing area). Under suitable hypotheses, he proved the existence of laminations and the persistence of foliations (similar to KAM theory).

**Positions.** Moser was a professor at the Courant Institute of Mathematical Science of New York University from 1960 to 1980, and director of the Courant Institute from 1967 to 1970. He was president of the International Mathematical Union from 1983 to 1986. He was director of the Research Institute for Mathematics of the Swiss Federal Institute of Technology from 1980 until his retirement in 1995.

He died of prostate cancer. He was survived by his wife, two daughters, a step-son, and six grandchildren.

## BIBLIOGRAPHY

Chern and Hirzebruch, eds., Wolf Prize in Mathematics,vol. 2 (cited below) contains a complete bibliography of Moser’s mathematical works.

### WORKS BY MOSER

“A New Technique for the Construction of Solutions of Nonlinear Differential Equations.” *Proceedings of the National Academy of Sciences of the United States of America* 47 (1961): 1824–1831.

“On Invariant Curves of Area-Preserving Mappings of an Annulus.” *Nachrichten Akademie Wissenschaften Göttingen Mathematische-Physikalische Klasse* 11 (1962): 1–20.

With William H. Jefferys. “Quasi-Periodic Solutions for the Three-Body Problem.’’ *Astronomical Journal* 71 (1966): 568–578.

“A Rapidly Convergent Iteration Method and Nonlinear Partial Differential Equations, I and II.” *Annali Scuola Normale Superiore Pisa* 20 (1966): 265–315, 499–535. The Nash-Moser theory.

“Lectures on Hamiltonian Systems.” *Memoirs of the American*

*Mathematical Society* 81 (1968): 1–60. Includes a discussion of a containment problem for magnetic field lines.

“Quasi-Periodic Solutions of the Three-Body Problem.’’ *Bulletin Astronomique* 3 (1968): 53–59.

*Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics*. Annals of Mathematics Studies 77. Princeton, NJ: Princeton University Press, 1973.

With Carl Ludwig Siegel. *Lectures on Celestial Mechanics*. Translation by Charles I. Kalme of *Die Grundlehren der Mathematische Wissenschaften* 187. New York and Heidelberg, Germany: Springer-Verlag, 1971.

### OTHER SOURCES

Chern, Shiing-Shen, and Friedrich Hirzebruch, eds. *Wolf Prize in Mathematics*. Vol. 2. River Edge, NJ: World Scientific Publishing, 2001.

Féjoz, Jacques. “Démonstration du ‘théorème d’Arnold’ sur la stabilité du système planétaire (d’après Herman).” *Ergodic Theory and Dynamical Systems* 24 (2004): 1521–1582.

Hasselblatt, Boris, and Anatole Katok. “The Development of Dynamics in the 20th Century and the Contribution of Jürgen Moser.” *Ergodic Theory and Dynamical Systems* 22 (2002): 1343–1364. A very extensive discussion of Moser’s mathematical work in dynamics.

Kolmogorov, Andrey N. “On the Conservation of Conditionally Periodic Motions for a Small Change in Hamilton’s Function.” *Doklady Academiia Nauk SSSR* 98 (1954): 527–530 (in Russian). English translation, Casati and Ford, eds., *Stochastic Behavior in Classical and Quantum Hamilton Systems* (Volta Memorial Conference Como, 1977). Lecture Notes in *Physics* 93 (1997): 51–56.

Lax, Peter D. “Jürgen Moser, 1928–1999.” *Ergodic Theory and Dynamical Systems* 22 (2002): 1337–1342. Moser’s life and personality by a close friend, together with an impressionistic appreciation of his work.

Mather, John N., Henry P. McKean, Louis Nirenberg, et al. “Jürgen K. Moser (1928–1999).” *Notices of the American Mathematical Society* 47 (December 2000): 1392–1405. A brief discussion of Moser’s life, together with an extensive discussion of his mathematical work and a bibliography.

Poincaré, Henri. “Sur le problème des trois corps et les équations de la dynamique.’’ *Acta Mathematica* 13 (1890): 1–227.

Siegel, Carl Ludwig. *Vorlesungen über Himmelsmechanik*. Berlin, Göttingen, and Heidelberg: Springer-Verlag, 1956.

*John N. Mather*

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**Moser, Jürgen K.**