Vladimir Igorevich Arnold, one of the great mathematicians of the 20th century, died unexpectedly in Paris on 3 June 2010. His achievements were remarkable both for their depth and breadth, and his work had and continues to have a significant impact on the formulation and solution of problems in science and engineering.
Arnold was born on 12 June 1937 in Odessa, Soviet Union. Exposed to mathematical problem solving at a young age, he became a student at Moscow State University, where he received his undergraduate degree in 1954 and his PhD in 1961. His mentor was the mathematician-physicist Andrei Kolmogorov. Arnold’s outstanding mathematical abilities became apparent early on when, at the age of 19, he solved Hilbert’s 13th problem, showing that seventh-order equations can be solved using functions of two variables. His subsequent work covered an extraordinary range of topics, in many cases opening up new areas of research in mathematics and physics.
In 1954 Kolmogorov proved the existence of quasi-periodic solutions in perturbations of integrable Hamiltonian systems. That problem, motivated by the question of the stability of the solar system, had vexed mathematicians for almost a century, and Kolmogorov’s approach provided a scheme for controlling the troublesome “small divisors.” Arnold developed a different approach that yielded deeper insights, and, at around the same time, Jürgen Moser enlarged the class of systems that could be treated. The resulting collection of mathematical theorems and techniques is now known as the Kolmogorov-Arnold-Moser (KAM) theory. The KAM theory continues to influence work on global Hamiltonian dynamics across a broad and growing range of applications, such as the foundations of classical statistical mechanics (see the article by Thierry Dauxois, Physics Today, January 2008, page 55), ionization of atoms in electromagnetic fields, the internal dynamics of molecules, and the theory of mixing in micro- and nanofluidics. Describing what are now known as Arnold tongues, Arnold also obtained seminal results on the dynamics of maps of the circle. That work provided a foundation for understanding the ubiquitous phenomenon of frequency locking in nonlinear systems.
In the early 1960s, Arnold constructed a Hamiltonian system that exhibited global instability. Cleverly stripping away technical complications, he demonstrated a general geometrical mechanism, now referred to as Arnold diffusion, for instability in systems with more than two degrees of freedom. Such global instabilities have been used to explain aspects of the dynamics of the solar system, the behavior of cold atoms, and energy transfer in molecular systems.
The KAM theory and Arnold diffusion were the cornerstones of a new vision for Hamiltonian dynamics; those ideas were taken up by Joe Ford, Boris Chirikov, and many others, and led to the rapid development of the fields of Hamiltonian chaos, both classical and quantum.
In the late 1960s, Arnold published several papers on classical hydrodynamic stability. His geometrical approach not only yielded new insights but also allowed generalization and improvement of classical stability results for Euler flows. The Arnold stability theorems and the Arnold method are now standard tools in hydrodynamics, and that work has led to the fertile field of topological fluid mechanics.
Arnold made equally fundamental contributions to the areas of singularity theory (the rigorous mathematics behind catastrophe theory), asymptotics (relevant to wave propagation, caustics, and quantum chaos), and symplectic topology.
Geometrical ideas pervade Arnold’s work, which perhaps explains the rapid application of his ideas to physical problems. His harsh criticisms of overly formalistic mathematics undoubtedly resonated with physical scientists, as did his famous aphorism, “Mathematics is the part of physics where experiments are cheap.”
Arnold was a prolific author of textbooks, problem books, and historical surveys. With André Avez, he wrote Ergodic Problems of Classical Mechanics (W. A. Benjamin, 1968). That seminal book, still one of the best introductions to dynamical systems theory, offered a balanced discussion of Hamiltonian dynamics and ergodic theory; it also popularized the chaotic torus map now known as the Arnold cat map. Mathematical Methods of Classical Mechanics (Springer), first published in 1978, was immediately recognized as a classic and hailed by reviewers as “a masterpiece” and “pure poetry.” The text gives a clear modern proof of the so-called Arnold-Liouville-Mineur theorem on the existence of action-angle variables and beautifully confirms the truth of Arnold’s dictum that “Hamiltonian mechanics cannot be understood without differential forms.” It has become essential reading for physicists and mathematicians alike. Arnold’s books on differential equations demonstrate the power of qualitative approaches in the spirit of Henri Poincaré, as opposed to traditional “cookbook” methods.
Like Kolmogorov, Arnold was deeply concerned with the mathematical curriculum. He had firm ideas about the way mathematics should be taught, and he was particularly critical of the axiomatization of mathematics for its own sake and the resulting divorce from the natural sciences. In 2005 he published Arnold’s Problems (Springer), a collection of problems posed in his widely renowned seminar, which ran for more than 30 years in Moscow. His many students have themselves had a major impact on the mathematical landscape.
Arnold received numerous awards, including the 1965 Lenin Prize, the 2001 Dannie Heineman Prize for Mathematical Physics, and the 2001 Wolf Prize in Mathematics. It is now known that pressure from the Soviet authorities prevented him from receiving the 1974 Fields Medal, one of the highest awards in mathematics.
Arnold was a professor in the faculty of mechanics and mathematics at Moscow State University from 1965 to 1986. In 1986 he moved to the Steklov Mathematical Institute in Moscow, and in 1993 he accepted a position at the Paris-Dauphine University; thereafter he divided his time between the Steklov Institute and Paris.
The career of Vladimir Arnold showed that despite increasing specialization in mathematics and science in general, it is still possible for one person to make contributions of the first rank in both pure and applied fields of mathematics as well as in the physical sciences. His sudden loss comes as a shock, but his legacy as a scholar and teacher will serve as an inspiration for many years to come.
