Mathematics of the Heavens Robert Osserman Sponsored each April by the Joint Policy Board for Mathematics, Mathematics Awareness Month provides an opportunity to celebrate mathematics and its uses. The theme for Mathematics Awareness Month 2005 is “Mathematics and the Cosmos”. In this article, the Notices reproduces three Mathematics Awareness Month “theme essays” written by Robert Osserman. Two other theme essays, plus a variety of resources including a Mathematics Awareness Month poster, are available on the website http://www.mathaware.org. by the publication of Laplace’s Méchanique Céleste in five volumes Mathematics from 1799 to 1825 and Poincaré’s Les Méthodes Nouvelles de la and the Cosmos Méchanique Céleste in three volumes from 1892 to 1899. The mid-nineteenth century produced further important contribu- Introduction and Brief History tions from Jacobi and Liouville, among others, as well as two The mathematical study of the cosmos has its roots groundbreaking new directions that were to provide the basic tools in antiquity with early attempts to describe the leading to the two revolutionary breakthroughs of twentieth-cen- motions of the Sun, Moon, stars, and planets in pre- tury physics; Hamilton’s original approach to dynamics became cise mathematical terms, allowing predictions of the springboard for quantum mechanics and the general subject future positions. In modern times many of the of dynamical systems, while Riemann’s 1854 “Habili- greatest mathematical scientists turned their at- tationsschrift” introduced curved spaces of three and more di- tention to the subject. Building on Kepler’s dis- mensions as well as the general notion of an n-dimensional man- covery of the three basic laws of planetary motion, ifold, thus ushering in the modern subject of cosmology leading Newton invented the subjects of “celestial me- to Einstein and beyond. chanics” and dynamics. He studied the “n-body The twentieth century saw a true flowering of the subject, as problem” of describing the motion of a number of new mathematical methods combined with new physics and masses, such as the Sun and the planets and their rapidly advancing technology. One might single out three main moons, under the force of mutual gravitational at- areas, with many overlaps and tendrils reaching out in multiple traction. He was able to derive improved versions directions. of Kepler’s laws, one of whose consequences yielded First, cosmology became ever more intertwined with astro- dramatic results just in the past decade when it was physics, as discoveries were made about the varieties of stars and used to detect the existence of planets circling their life histories, as well as supernovae and an assortment of other stars. previously unknown celestial objects, such as pulsars, quasars, The two leading mathematicians of the eigh- dark matter, black holes, and even galaxies themselves, whose ex- teenth century, Euler and Lagrange, both made istence had been suspected but not confirmed until the twenti- fundamental contributions to the subject, as did eth century. Most critical was the discovery at the beginning of Gauss at the turn of the century, spurred by the dis- the century of the expansion of the universe, with its concomi- covery of the first of the asteroids, Ceres, on Jan- tant phenomenon of the Big Bang, and then at the end, the recent uary 1, 1801. The nineteenth century was framed discovery of the accelerating universe, with its associated con- jectural “dark energy”. Robert Osserman is the special projects director of the Second, the subject of celestial mechanics evolved into that of Mathematical Sciences Research Institute in Berkeley. He dynamical systems, with major advances by mathematicians also serves as chair of the Mathematics and the Cosmos Advisory Committee for Mathematics Awareness Month G. D. Birkhoff, Kolmogorov, Arnold, and Moser. Many new dis- 2005. His email address is [email protected]. coveries were made about the n-body problem, both general ones, The author gratefully acknowledges the input of Myles such as theorems on stability and instability, and specific ones, such Standish, Martin Lo, Fabrizio Polara, Susan Lavoie, and as new concrete solutions for small values of n. Methods of chaos Charles Avis of the Jet Propulsion Laboratory, and Jerry theory began to play a role, and the theoretical studies were both Marsden of Caltech. informed by and applied to the profusion of new discoveries of APRIL 2005 NOTICES OF THE AMS 417 planets, their moons, asteroids, comets, and other rockets, artificial satellites, and space probes. On objects composing the increasingly complex struc- the other hand, almost all of those space vehicles ture of the solar system. were equipped with scientific instruments for gath- Third, the advent of actual space exploration, ering data about the Earth and other objects in our sending artificial satellites and space probes to solar system, as well as distant stars and galaxies the furthest reaches of the solar system, as well as going back to the cosmic microwave background the astronaut and cosmonaut programs for nearby radiation. Furthermore, the deviations in the paths study, transformed our understanding of the ob- of satellites and probes provide direct feedback on jects in our solar system and of cosmology as a the gravitational field around the Earth and whole. The Hubble space telescope was just one of throughout the solar system. many viewing devices, operating at all wave lengths, Beyond these direct effects, there are many other that provided stunning images of celestial objects, areas of interaction between the space program and near and far. The 250-year-old theoretical discov- mathematics. We list just a few: eries by Euler and Lagrange of critical points known • GPS: the global positioning system, as “Lagrange points” saw their practical application • data compression techniques for transmitting in the stationing of satellites. The 100-year-old messages, introduction by Poincaré of stable and unstable • digitizing and coding of images, manifolds formed the basis of the rescue of • error-correcting codes for accurate transmis- otherwise abandoned satellites, as well as the plan- sions, ning of remarkably fuel-efficient trajectories. • “slingshot” or “gravitational boosting” for Finally, the biggest twentieth-century innova- optimal trajectories, tion of all, the modern computer, played an ever- • exploitation of Lagrange points for strategic increasing and more critical role in all of these ad- placement of satellites, vances. Numerical methods were applied to all • dynamical systems methods for energy- three of the above areas, while simulations and com- efficient orbit placing, puter graphics grew into a major tool in deepen- • finite element modeling for structures such as ing our understanding. The ever-increasing speed spacecraft and antennas. and power of computers went hand-in-hand with Some of the satellites and space probes that the increasingly sophisticated mathematical meth- have contributed to cosmology and astrophysics are ods used to code, compress, and transmit messages • the Hubble space telescope, and images from satellites and space probes span- • the Hipparcos mission to catalog the posi- ning the entire breadth of our solar system. tions of a million stars to new levels of accuracy, • the COBE and WMAP satellites for studying the General References cosmic microwave background radiation, [1] DENNIS RICHARD DANIELSON, The Book of the Cosmos: • the Genesis mission and SOHO satellite for Imagining the Universe from Heraclitus to Hawking, studying the Sun and solar radiation, Perseus, Cambridge, 2000. the ISEE3/ICE space probe to study solar flares [2] BRIAN GREENE, The Fabric of the Cosmos. Space, Time, • and the Texture of Reality, Alfred A. Knopf, Inc., New and cosmic gamma rays before going on to visit the York, 2004. MR2053394. Giacobini-Zimmer comet and Halley’s comet, [3] THOMAS S. KUHN, The Copernican Revolution: Planetary • the LAGEOS satellites to test Einstein’s pre- Astronomy in the Development of Western Thought, diction of “frame dragging” around a rotating body. Harvard University Press, Cambridge, 1957. Rather than trying to cover all or even most of [4] ROBERT OSSERMAN, Poetry of the Universe: A Mathemat- the mathematical links, we focus on two that ical Exploration of the Cosmos, Anchor/Doubleday, are absolutely essential and central to the whole New York, 1995. endeavor: first, navigation and the planning of [5] GEORGE PÓLYA, Mathematical Methods in Science, rev. ed. trajectories; and second, communication and the (Leon Bowden, ed.), New Mathematical Library, Vol. 26. Mathematical Association of America, Washington, transmission of images. DC, 1977. MR0439511 (55:12401) [6] STEVEN WEINBERG, The First Three Minutes, A Modern View Navigation, Trajectories, and Orbits of the Origin of the Universe, 2nd ed., Basic Books, 1993. When the U.S. space program was set up in earnest—a process described in detail in the recent History Channel documentary Race to the Moon— Space Exploration a notable feature was the introduction of the Starting in the twentieth century, the mathemati- Mission Control Center. The first row of seats in cal exploration of the cosmos became inextricably mission control was known as “the trench”, and it entwined with the physical exploration of space. On is from there that the mathematicians whose spe- one hand, virtually all the methods of celestial me- cialty is orbital mechanics kept track of trajecto- chanics that had been developed over the centuries ries and fed in the information needed for were transformed into tools for the navigation of navigation. Their role is particularly important for 418 NOTICES OF THE AMS VOLUME 52, NUMBER 4 operations involving rendezvous between two vehicles, in delicate operations such as landings on the Moon and in emergencies that call on all their skills, the most notable of which was bringing back alive the crew of Apollo 13 after they had to aban- don the command module and were forced to use the lunar landing module—never designed for that purpose—to navigate back to Earth.