Improbability of Collisions in n-Body Systems

  • Stefan Fleischer
  • Andreas KnaufEmail author


For a wide class of two-body interactions, including standard examples like gravitational or Coulomb fields, we show that collision orbits in n-body systems are of Liouville measure zero for all energies. We use techniques from symplectic geometry to relate the volume of the union of collision orbits to the area of Poincaré surfaces surrounding the collision set.



We thank the anonymous reviewer for his/her helpful comments.


  1. 1.
    Dereziński, J., Gérard, C.: Scattering theory of classical and quantum N-particle systems. Texts and Monographs in Physics. Springer, Berlin (1997)Google Scholar
  2. 2.
    Fleischer , S., Knauf, A.: Improbability of wandering orbits passing through a sequence of poincaré surfaces of decreasing size. Arch. Ration. Mech. Anal. 231(3), 1781–1800, 2019MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Knauf, A.: Mathematical Physics: Classical Mechanics. Springer, New York 2018CrossRefzbMATHGoogle Scholar
  4. 4.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford University Press, Oxford 1999zbMATHGoogle Scholar
  5. 5.
    Pollard, H., Saari, D.G.: Singularities of the \(n\)-body problem I. Arch. Ration. Mech. Anal. 30, 263–269, 1968MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Saari, D.G.: Improbability of collisions in Newtonian gravitational systems. Trans. Amer. Math. Soc., 162, 267–271; erratum, ibid. 168(1972), 521, 1971Google Scholar
  7. 7.
    Saari , D.G.: Improbability of collisions in Newtonian gravitational systems II. Trans. Am. Math. Soc. 181, 351–368, 1973MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sullivan, J.M.: Curves of finite total curvature. Discrete Differ. Geom. 38, 137–161, 2008MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsFriedrich-Alexander-University Erlangen-NürnbergErlangenGermany

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