Singularity theory predicts that under certain circumstances slow, smooth changes in a system can lead to an abrupt major change, in the way that the slipping of a few small rocks can set off an avalanche. The theory has applications in physics, chemistry and biology.
“He was a genius and one of the greatest and most influential mathematicians of our time,” said Boris A. Khesin, a former student of Dr. Arnold’s and now a professor of mathematics at the University of Toronto.
One of Dr. Arnold’s biggest contributions was applying the methods of geometry and symmetry to the motion of particles. Dr. Arnold work on how fluids flow was applied to the dynamics of weather, providing a mathematical explanation for why it is not possible to make forecasts months in advance. Infinitesimal gaps or errors in information cause forecasts to diverge completely from reality.
A similar approach can also be applied to the motion of planets. If Earth were the only planet to circle the Sun, its orbit would follow a precise elliptical path, but the gravity of the other planets disturbs the motion. Scientists found that it impossible to calculate the precise motion of the planets over very long periods of time or even prove that Earth will not one day be flung out of the solar system.
Understanding the subtle and difficult-to-predict boundary between stability and instability is important not only in the study of planetary dynamics but also in other endeavors, like designing a nuclear fusion reactor.
In 1954, the Russian mathematician Andrey Kolmogorov figured out a key insight to calculating whether such systems are stable. Dr. Arnold provided a rigorous proof in 1963 for one set of circumstances. Another mathematician, Jürgen Moser, provided the proof for another. The work is now collectively know at the KAM theory.