## Mathematics and the Cosmos

### Introduction and Brief History

The mathematical study of the cosmos has its roots in antiquity with early attempts to describe the motions of the sun, moon, stars and planets in precise mathematical terms, allowing predictions of future positions. In modern times many of the greatest mathematical scientists turned their attention to the subject. Building on Kepler's discovery of the three basic laws of planetary motion, Newton invented the subjects of celestial mechanics and dynamics. He studied the "n-body problem" of describing the motion of a number of masses, such as the sun, planets and their moons, under the force of mutual gravitational attraction. He was able to derive improved versions of Kepler's laws, one of whose consequences yielded dramatic results just in the past decade when it was used to detect the existence of planets circling other stars.

The two leading mathematicians
of the 18^{th} century, Euler and Lagrange, both
made fundamental contributions to the subject, as did
Gauss at the turn of the century, spurred by the discovery
of the first of the asteroids, Ceres, on January 1, 1801.
The 19^{th} century was framed by the publication
of Laplace's *Méchanique Céleste * in
five volumes from 1799 to 1825, and Poincaré's *Les
Méthodes Nouvelles de la Méchanique Céleste * in
three volumes from 1892 to 1899. Mid-19^{th} century
produced further important contributions from Jacobi
and Liouville, among others, as well as two ground-breaking
new directions that were to provide the basic tools leading
to the two revolutionary breakthroughs of 20^{th} century
physics; Hamilton's original approach to dynamics became
the springboard for quantum mechanics and the general
subject of dynamical systems, while Riemann's 1854 *Habilitations * Lecture
introduced curved spaces of three and more dimensions
as well as the general notion of an n-dimensional manifold,
thus ushering in the modern subject of cosmology leading
to Einstein and beyond.

The 20^{th} century saw
a true flowering of the subject, as new mathematical
methods combined with new physics and rapidly advancing
technology. One might single out three main areas, with
many overlaps and tendrils reaching out in multiple directions.

First, cosmology became ever
more intertwined with astrophysics, as discoveries were
made about the varieties of stars and their life histories,
as well as supernovae, and an assortment of previously
unknown celestial objects, such as pulsars, quasars,
dark matter, black holes, and even galaxies themselves,
whose existence had been suspected but not confirmed
until the 20^{th} century. Most critical was
the discovery at the beginning of the century of the
expansion of the universe, with its concomitant phenomenon
of the Big Bang, and then at the end, the recent discovery
of the accelerating universe, with its associated conjectural "dark
energy."

Second, the subject of celestial mechanics evolved into that of dynamical systems, with major advances by mathematicians G.D. Birkhoff, Kolmogorov, Arnold, and Moser. Many new discoveries were made about the n-body problem, both general ones, such as theorems on stability and instability, and specific ones, such as new concrete solutions for small values of n. Methods of chaos theory began to play a role, and the theoretical studies were both informed by and applied to the profusion of new discoveries of planets, their moons, asteroids, comets and other objects composing the increasingly complex structure of the solar system.

Third, the advent of actual space exploration, sending artificial satellites and space probes to the furthest reaches of the solar system, as well as the astronaut and cosmonaut programs for nearby study, transformed our understanding of the objects in our solar system and of cosmology as a whole. The Hubble Space Telescope was just one of many viewing devices, operating at all wave lengths, that provided stunning images of celestial objects, near and far, The 250 year-old theoretical discoveries by Euler and Lagrange of critical points known as "Lagrange points" saw their practical application in the stationing of satellites. The 100 year-old introduction by Poincaré of stable and unstable manifolds formed the basis of the rescue of otherwise abandoned satellites, as well as the planning of remarkably fuel-efficient trajectories.

Finally, the biggest 20^{th} century
innovation of all, the modern computer, played an ever
increasing and more critical role in all of these advances.
Numerical methods were applied to all three of the above
areas, while simulations and computer graphics grew to
a major tool in deepening our understanding. The ever-increasing
speed and power of computers went hand-in-hand with the
increasingly sophisticated mathematical methods used
to code, compress, and transmit messages and images from
satellites and space probes spanning the entire breadth
of our solar system.

Robert
Osserman

(Chair), Special Projects Director,

Mathematical Sciences Research Institute

Email: ro
at msri.org

### General References

- Dennis Richard Danielson,
*The Book of the Cosmos: Imagining the Universe from Heraclitus to Hawking*, Cambridge, Perseus, 2000. - Brian Greene, The fabric of the cosmos. Space, time, and the texture of reality. Alfred A. Knopf, Inc., New York, 2004. xii+573 pp. ISBN: 0-375-41288-3 MR2053394
- Thomas S. Kuhn,
*The Copernican Revolution: Planetary Astronomy in the Development of Western Thought*, Cambridge, Harvard University Press, 1957. - Robert Osserman,
*Poetry of the Universe: A Mathematical Exploration of the Cosmos,*New York, Anchor/Doubleday 1995. - George Pólya, Mathematical methods in science. Revised edition. Edited by Leon Bowden. New Mathematical Library, Vol. 26. The Mathematical Association of America, Washington, D. C., 1977. xi+234 pp. MR0439511