Except for the painful beginning and end, well-deserved good fortune accompanied Moser all his life. Accomplishment and recognition came to him early and lastingly. He was the first recipient of the G. D. Birkhoff Prize of the AMS and SIAM. He was honored by a Wolf Prize, medals were showered on him, academies vied to elect him as their member. He was an outstanding expositor, much in demand. He delivered the Gibbs lecture of the AMS, the Pauli lectures of the ETH, the Hardy lectures of Cambridge University, the Hermann Weyl lectures of the Accademia Lincei, and the von Neumann lecture of SIAM. He was president of the IMU for three years.

He was fortunate in having many good students, with whom he continued close scientific and personal relations. He loved his family fiercely, two daughters and a stepson, six grandchildren and charming and talented sons-in-law.

According to an old Roman proverb, ‘what doesn't destroy me makes me stronger’. This fits Jürgen, for in his early youth he was exposed to deadly danger, but survived the stronger for it. In 1938, the ten year old Jürgen was offered a chance to enroll in an elite boarding school for future leaders, but his parents who saw the Nazi's plan were able to extricate him. Jürgen's father was a neurologist, and as such was sometimes called upon to declare someone as ‘unfit to live’ according to the Nazi eugenics program. Jürgen's father, courageously, always refused to issue such certifications.

Jürgen Moser (1928–1999) was one of the most accomplished mathematicians of the second half of the 20th century and his work had a major impact in broad areas of analysis, especially partial differential equations and dynamical systems, and geometry. In this article we discuss only his contributions to dynamics and closely related areas. We feel that the best way to make the reader understand and appreciate the impact of Moser's work is to discuss it within the framework of some major trends in the development of dynamical systems during the last century. We apologize in advance for omitting many important references, including some key original work. These can be found in secondary sources to which we refer for detailed accounts of various areas. Needless to say, we included in our references all papers by Moser relevant to our discussion. For brief surveys of Moser's work in all areas see [45,91].

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

The finite Pfaff lattice is given by a commuting Lax pair involving a finite matrix L (zero above the first subdiagonal) and a projection onto sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t_1,t_2,\dotsc, after conjugation by a diagonal matrix. The sequence of polynomial \tau-functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of \tau-functions is given inductively by the action of a fixed vertex operator.

As an example, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.

Minimization arguments are applied to a renormalized length functional to establish the existence of heteroclinic, homoclinic and chaotic geodesics on a two-dimensional torus satisfying some geometrical conditions.

Analytic quasi-periodic skew-systems with close to constant coefficients and with Diophantine frequencies seem to be, in a measure theoretical sense, ‘generically’ reducible. In this paper we show that such systems are, in a topological sense, generically uniquely ergodic and, in particular, non-reducible.

\widetilde{J}-holomorphic cylinders in \mathbb{R}\times M, where M is a contact manifold are investigated. It is shown that a sufficiently long cylinder having small area is close to a constant map, if its center action vanishes. If its center action is positive, it is close to a cylinder over a periodic orbit of the Reeb vector field, and has a well-determined shape. A compactness result is deduced, which is useful in symplectic field theory.

In this paper, we continue our investigation of dissipative PDE's forced by random bounded kick-forces and of the corresponding random dynamical system (RDS) in function spaces. It has been proved that the domain \mathcal{A} of attainability from zero (which is a compact subset of a function space) is invariant for the RDS associated with the original equation and carries a stationary measure \mu, which is unique among all measures supported by \mathcal{A}. Here we show that \mu is the unique stationary measure for the RDS in the whole space and study its ergodic properties.

In this paper, we point out a close relationship between two standard classical problems in mechanics which have coexisted in textbooks for many decades: (1) the pendulum whose suspension point executes fast periodic motion along a given curve; and (2) the skate (known also as the Prytz planimeter, or the ‘bicycle’). More generally, we deal with dynamical systems subjected to rapidly oscillating forcing. Examples include: charged particles in rapidly oscillating electromagnetic fields, in particular the Paul trap; particles in an acoustic wave; a bead sliding on a rapidly vibrating hoop. It turns out that the averaged systems of such kind are approximated by a non-holonomic system. The holonomy turns out to have a transparent geometrical or physical interpretation. For the example of a particle in an acoustic wave the holonomy is directly proportional to the speed of the vibration-induced drift.

Let X be a compact, connected subset of a smooth manifold. Suppose that X admits a codimension d lamination, d\ge 2. Let x,y\in X and \epsilon>0. There exists a sequence x=x_0,\dotsc,x_k=y in X such that the sum of the dth powers of the distances between successive points is less than \epsilon. We discuss two proposed applications of this result.

Moser's celebrated theorem guarantees that every diffeomorphism of a closed manifold can be isotoped to a volume-preserving one. We show that this statement cannot be extended into the contact category: some connected components of contactomorphism groups contain no volume-preserving maps. Thus, the dissipation of volume appears for purely topological reasons. This phenomenon can be considered from different viewpoints: geometric (isometric action of the contact mapping class group on the moduli space of contact forms), topological (action in symplectic homology) and dynamical (propagation of trajectories for symplectic maps). We define a numerical invariant—a contact Lyapunov exponent—which leads to a quantitive version of the above-mentioned result.

We describe the construction of almost-periodic solutions for a particular Schrödinger equation on a finite x- interval, depending on some potential V. The construction proceeds by iterating a KAM theorem about the existence of quasi-periodic solutions for some partial differential equations. As a result, we find for ‘almost all’ potentials V uncountably many almost-periodic solutions accumulating at the zero solution.

We show the stability of an elliptic fixed point of an analytic area-preserving mapping under the Bruno condition, using Moser's Twist-mapping Theorem. We require that there are non-vanishing Birkhoff constants and prove the convergence of the transformation to the Birkhoff normal form if all Birkhoff constants are zero. The convergence theorem is formulated and proved for arbitrary dimensions.

An integrable discrete-time Lagrangian system on the group of area-preserving plane diffeomorphisms \mathit{SDiff}(\mathbb{R}^{2}) is considered. It is shown that non-trivial dynamics exists only for special initial data and the corresponding mapping can be interpreted as a Bäcklund transformation for the (simple) Monge–Ampère equation. In the continuous limit, this gives the isobaric (constant pressure) solutions of the Euler equations for an ideal fluid in two dimensions. In the Appendix written by E. V. Ferapontov and A. P. Veselov, it is shown how the discrete system can be linearized using the transformation of the simple Monge–Ampère equation going back to Goursat.