Math Unites the Celestial and the Atomic
In recent years, researchers have developed astonishing new insights into a hidden unity between the motion of objects in space and that of the smallest particles. It turns
out there is an almost perfect parallel between the mathematics describing celestial mechanics and the mathematics governing some aspects of atomic physics. These insights have led to new ways to design space missions, as described in the article, “Ground Control to Niels Bohr: Exploring Outer Space with Atomic Physics” by Mason Porter and Predrag Cvitanovic, which appears in the October 2005 issue of the Notices of the American Mathematical Society. The article describes work by, among other scientists, physicist Turgay Uzer of the Georgia Institute of Technology, mathematician Jerrold Marsden of the California Institute of Technology and engineer Shane Ross of the University of Southern California. Imagine a group of celestial bodies — say, the Sun, the Earth, and a Spacecraft — moving along paths determined by their mutual gravitational attraction. The mathematical theory of dynamical systems describes how the bodies move in relation to one another. In such a celestial system, the tangle of gravitational forces creates tubular “highways” in the space between the 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 of chemical 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, the article states, “The orbits used to design space missions 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.