Remote access

How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax

Relative equilibria in the 3-dimensional curved $n$-body problem


About this Title

Florin Diacu

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 228, Number 1071
ISBNs: 978-0-8218-9136-0 (print); 978-1-4704-1483-2 (online)
DOI: http://dx.doi.org/10.1090/memo/1071
Published electronically: July 22, 2013
Keywords:Celestial mechanics, $n$-body problems, spaces of constant curvature, fixed points, periodic orbits, quasiperiodic orbits, relative equilibria

View full volume PDF

View other years and numbers:

Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. BACKGROUND AND EQUATIONS OF MOTION
  • Chapter 3. ISOMETRIES AND RELATIVE EQUILIBRIA
  • Chapter 4. CRITERIA AND QUALITATIVE BEHAVIOUR
  • Chapter 5. EXAMPLES
  • Chapter 6. CONCLUSIONS

Abstract


We consider the -dimensional gravitational -body problem, , in spaces of constant Gaussian curvature , i.e. on spheres , for , and on hyperbolic manifolds , for . Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. We also briefly discuss the issue of singularities in order to avoid impossible configurations. We derive the equations of motion and define six classes of relative equilibria, which follow naturally from the geometric properties of and . Then we prove several criteria, each expressing the conditions for the existence of a certain class of relative equilibria, some of which have a simple rotation, whereas others perform a double rotation, and we describe their qualitative behaviour. In particular, we show that in the bodies move either on circles or on Clifford tori, whereas in they move either on circles or on hyperbolic cylinders. Then we construct concrete examples for each class of relative equilibria previously described, thus proving that these classes are not empty. We put into the evidence some surprising orbits, such as those for which a group of bodies stays fixed on a great circle of a great sphere of , while the other bodies rotate uniformly on a complementary great circle of another great sphere, as well as a large class of quasiperiodic relative equilibria, the first such non-periodic orbits ever found in a 3-dimensional -body problem. Finally, we briefly discuss other research directions and the future perspectives in the light of the results we present here.

References [Enhancements On Off] (What's this?)


Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia