Singularities of the N-Body Problem


PM014 186 pages ISBN 2-921120-20-8 1992

Florin N. Diacu

The text is based on a series of lectures given at the CRM in 1991. It contains many recent results as well as old ones regarded from a modern point of view. The main purpose is the study of the singularities arising in the equations of motion and in the solutions of the classical N-body problem of celestial mechanics, using tools of dynamical systems theory. Topics include criteria for existence of solutions with singularities like the theorems of von Zeipel, Pollard and Saari, examples of solutions leading to pseudocollisions, an introduction to the growing theory of central configurations, regularization techniques for collisions, McGehee transformations and the idea of the collision manifold, structural stability of homothetic orbits, simultaneously binary collisions and series solutions attempts for the N-body problem. The text offers a modern introduction to celestial mechanics from a mathematical point of view.

Contents

  1. The N-body problem
  2. The nature of singularities
  3. The theorem of von Zeipel
  4. The criteria of Pollard and Saari
  5. Solutions free of pseudocollisions
  6. Examples of pseudocollisions
  7. Central configurations
  8. Regularization of collisions
  9. Triple collision in the rectilinear three-body problem
  10. The collision manifold
  11. Structural stability of homothetic orbits
  12. Simultaneous binary collisions
  13. Time regularization of partial collisions
  14. The solution of the N-body problem

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