Fifteen Eighty Four

Despite having been alive for more than three centuries the classical N-body problem remainsalive and well! In this book I demonstrate its vibrancy by exploring four open questions within the problem.

The book was born during the beginning of the pandemic when Marcelo Disconzi asked me to give a
Zoom colloquium talk at Vanderbilt University, which I did. The body of the book consists of four chapters, a chapter per question. Preceding the questions are two chapters, one of which tours some of the known solutions and the second of which sets up the problem along with its symmetries, notation and mathematical structures to be used in the questions chapters. After the questions chapters are a number of appendices which help make the book more self-contained. The first appendix is a lightening overview of geometric mechanics.

The first question asks: “Is the set of central configurations a finite set?” and made it onto Smale’s list
of questions for mathematics in the 21st century. The central configurations are the only explicitly known
solutions to the problem and play a central role in a number of sub-areas of the problem. In the 1770s
Lagrange and Euler found all central configurations for N = 3. There are five. Over two hundred years
later, in 2006, Hampton and Moeckel proved the list of central configurations is finite when N = 4. The
problem was ‘almost” solved for N = 5 in 2012 by Albouy and Kaloshin and appears wide open for N = 6.

The oldest question posed is the 2nd: “Are there any stable periodic solutions for more than two bodies?”
Michel Hermann called a strengthened version of this question “the oldest question in dynamics” in his ICM lectures. You might think the existence of our solar system is proof that the answer is ‘yes’. It is not! Surprisingly, in 2020 Albouy and Dullin proved that if we allow our three bodies to move in four dimensions with full rank four dimensional angular momentum then the answer is provably ‘yes’. But for three bodies moving in three-space or the plane this question is utterly open. The KAM [Kolmogorov-Arnol’d-Moser] theorem lives in basin of attraction of this question. Having nowhere else to go, we delve into KAM, and Arnol’d diffusion, particularly regarding their relevance to the question.

I did no work myself on the first two questions. But I have made contributions to the last two questions.
The 3rd question asks ‘Is every braid type realized by a solution to the planar N-body problem?” and
has driven much of my research for the last 27 years. This question led to my rediscovery, with Alain
Chenciner, of the figure eight orbit, a pretty solution first found by Cris Moore. Our paper is referenced
on page 199 of the English translation of Cixin Li’s science fiction novel “The Three-Body Problem”, an
appearance which has given me the pleasure of having friends and miscellaneous kayakers occasionally write to me and ask if that really is my name appearing in the best-seller. The figure eight set off a
floodgate of discoveries of beautiful highly symmetric ‘choreography solutions’ and some of this work is surveyed.

The 4th question asks “Does the scattering map have an open dense image?” This question is partially
inspired by Rutherford scattering {the essential theoretical underpinnings for the discovery of the nucleus
at the center of each atom. Rutherford solved for the scattering map in the case N = 2. Unlike the other
chapters, this chapter concerns only positive energy solutions. Such solutions are always unbounded which allows us to speak about “scattering”. Setting up the scattering map properly for N > 2 takes some work. Andreas Knauf posed this question to me in 2012.

Title: Four Open Questions for the N-Body Problem

Author: Richard Montgomery

ISBN: 9781009200585