Modeling Life

Abstract

The text follows what we consider to be a twentieth-century math approach to the subject. The technical development of calculus in the eighteenth and nineteenth centuries saw differential equations as pieces of language, which were then to be operated on by paper-and-pencil tech- niques to produce other pieces of language (the “solutions”). This had worked well for Newton in the gravitational 2-body problem (1687), and was the paradigm for applied math in the centuries that followed. The Newtonian program came to a dramatic dead end with the 3-body problem, an obvious and more valid extension of the 2-body problem. The 3-body problem had proved analytically intractable for centuries, and in the late nineteenth century, results by Haretu and Poincaré showed that the series expansions that were the standard technique actually diverged. Then the discovery by Bruns that no quantitative methods other than series expansions could resolve the n-body problem meant the end of the line for the Newtonian program of writing a differential equation and solving it (Abraham and Marsden, 1978). It was Poincaré’s genius to see that while this represented “calculus : fail,” it was also the springboard for an entirely new approach that focused on topology and geometry and less on analytical methods. His groundbreaking paper was called “On the curves defined by a differential equation,” linking two very different areas: differential equations (language) and curves, which are geometrical objects. The distinction is critical: solution curves almost always exist (Picard– Lindelöf theorem), but their equations almost never do.