Vladimir Arnold

Vladimir Arnold (1937–2010) was a Soviet and Russian mathematician whose work shaped dynamical systems, singularity theory, classical mechanics, and symplectic geometry.

Dynamical system: A system whose state changes over time according to a fixed rule. Arnold studied how such systems evolve, especially when their behavior is nonlinear.
Arnold diffusion: Slow drift through action space in nearly integrable Hamiltonian systems with at least three degrees of freedom. Arnold proposed this mechanism as a way small perturbations can produce large long-term changes.
Andrey Kolmogorov: A Soviet mathematician who made foundational contributions to probability, turbulence, and dynamical systems. Kolmogorov was Arnold’s doctoral adviser and shared the KAM theorem’s authorship.
Celestial mechanics: The study of the motion of astronomical bodies under gravity. Arnold used perturbation theory to address the stability of planetary motion.
Hamiltonian mechanics: A formulation of classical mechanics using a function called the Hamiltonian to describe a system’s evolution. Arnold used its geometric structure to investigate stability and long-term motion.
Arnold conjecture: A family of conjectures relating fixed points of Hamiltonian diffeomorphisms to the topology of the underlying manifold. Its symplectic fixed-point questions grew from Arnold’s study of periodic orbits.
Jürgen Moser: A German-American mathematician known for work in analysis, dynamical systems, and partial differential equations. Moser independently extended the persistence results that joined his name to the KAM theorem.
KAM theory: A theory describing the persistence of quasiperiodic motion in nearly integrable Hamiltonian systems. Its results explain why some planetary trajectories remain orderly despite perturbations.
Symplectic geometry: The study of manifolds equipped with a closed, nondegenerate differential two-form. Arnold helped establish this geometry as a central language for classical mechanics.
Arnold's cat map: A hyperbolic torus automorphism that stretches and folds the torus while preserving area. This explicit model illustrates the mixing and sensitivity central to dynamical systems.

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