Monday, November 13, 2006

The Celestial & The Atomic

There is an almost perfect parallel between math describing the motion of celestial objects, like the sun (shown here in an ultraviolet image), and atomic objects.

Image courtesy of NASA

Imagine a group of celestial bodies say, the Sun, the Earth, and a Space craft moving along paths determined by their mutual gravitational attraction.

The mathematical theory of dynamical systems describes how the bodies movein relation to one another. In such a celestial system, the tangle of gravitational forces creates tubular "highways" in the space betweenthe bodies. If the spacecraft enters one of the highways, it is whisked along without the need to use very much energy. With help from mathematicians, engineers and physicists, the designers of the Genesis spacecraft mission used such highways to propel the craft to its destinations with minimal use of fuel.

In a surprising twist, it turns out that some of the same phenomena occur on the smaller, atomic scale. This can be quantified in the study of what are known as "transition states", which were first employed in the field of chemical dynamics. One can imagine transition states as barriers that need to be crossed in order for chemical reactions to occur (for"reactants" to be turned into "products"). Understanding the geometry of these barriers provides insights not only into the nature ofchemical reactions but also into the shape of the "highways" in celestial systems.

The connection between atomic and celestial dynamics arises because the same equations govern the movement of bodies in celestial systems and the energy levels of electrons in simple systems and these equations are believed to apply to more complex molecular systems as well.

This similarity carries over to the problems' transition states; the difference is that which constitutes a "reactant" and a "product" is interpreted differently in the two applications. The presence of the same underlying mathematical description is what unifies these concepts.

Because of this unifying description, it can be said that: "The orbits used to design spacemissions thus also determine the ionization rates of atoms and chemical-reaction rates of molecules!"

The mathematics that unites these two very different kinds of problems is not only of great theoretical interest for mathematicians, physicists, and chemists, but also has practical engineering value in space mission design and chemistry.
More from Science Daily releases 5th October 2005

Dynamic Systems Theory:
The Lorenz attractor &
Gauss' Modular flow

Image by
Etienne Ghys/Jos Leys

A collaboration between a mathematician and an artist-geometer has resulted in some of the most mathematically sophisticated and aesthetically gripping animations ever seen in the field. Their visualizations of cutting-edge research in dynamical systems theory not only provide a dramatic new way of visiting mathematical worlds once seen only in the mind's eye, but also point to a new era for the use of computer graphics in communicating and carrying out mathematical research.

In 1963, the meteorologist Edward Lorenz was studying a very simplified numerical model for the atmosphere, which led him to the amazing strange attractor popularized through the famous butterfly effect: the flapping wings of a butterfly might cause some tiny change in the state of the atmosphere which can in turn lead to hurricanes!
"We would like to describe a close topological connection between these two mathematical objects."
Feature column by Ghys & Leys
More from Science Daily releases 13th Nov 2006
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