Math Unites The Celestial And The Atomic

Thearticle describes work by, among other scientists, physicist TurgayUzer of the Georgia Institute of Technology, mathematician JerroldMarsden of the California Institute of Technology and engineer ShaneRoss of the University of Southern California.

Imagine a group ofcelestial bodies—say, the Sun, the Earth, and a Spacecraft—moving alongpaths determined by their mutual gravitational attraction. Themathematical theory of dynamical systems describes how the bodies movein relation to one another. In such a celestial system, the tangle ofgravitational forces creates tubular “highways” in the space betweenthe bodies. If the spacecraft enters one of the highways, it is whiskedalong without the need to use very much energy. With help frommathematicians, engineers and physicists, the designers of the Genesisspacecraft mission used such highways to propel the craft to itsdestinations with minimal use of fuel.

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

The connection between atomic and celestialdynamics arises because the same equations govern the movement ofbodies in celestial systems and the energy levels of electrons insimple systems—and these equations are believed to apply to morecomplex molecular systems as well. This similarity carries over to theproblems’ transition states; the difference is that which constitutes a“reactant” and a “product” is interpreted differently in the twoapplications. The presence of the same underlying mathematicaldescription is what unifies these concepts. Because of this unifyingdescription, the article states, “The orbits used to design spacemissions thus also determine the ionization rates of atoms andchemical-reaction rates of molecules!” The mathematics that unitesthese two very different kinds of problems is not only of greattheoretical interest for mathematicians, physicists, and chemists, butalso has practical engineering value in space mission design andchemistry.
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