Classical N-Body Hamiltonians

  • Jan Dereziński
  • Christian Gérard
Part of the Texts and Monographs in Physics book series (TMP)


A system of N non-relativistic particles moving in Euclidean space ℝ v is described with phase space ℝ Nv × ℝ Nv , with the coordinates
$$(x{}_1,...,{x_N},{\xi _1},...,{\xi _N})$$
, where (x i , ξ i ) are the position and momentum of the i-th particle. Its motion is described by the Hamiltonian
$$H(x,\xi ) = \sum\limits_{i = 1}^N {\frac{{\xi _i^2}}{{2{m_i}}}} + \sum\limits_{i < j} {{V_{ij}}} V({x_i} - {x_j})$$
, where m i is the mass of the i-th particle and V ij(x) is the interaction potential between particles i and j. The most important case of such a Hamiltonian is the one encountered in celestial mechanics where11111Typical assumptions that we will keep in mind in this chapter are
$${V_{ij}}(x) = \frac{{ - {m_i}{m_j}}}{{\left| x \right|}}$$




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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Jan Dereziński
    • 1
  • Christian Gérard
    • 2
  1. 1.Department of Mathematical Methods in PhysicsWarsaw UniversityWarsawPoland
  2. 2.Centre de MathématiquesEcole PolytechniquePalaiseau CedexFrance

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