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Classical N-Body Hamiltonians

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Abstract

A system of N non-relativistic particles moving in Euclidean space ℝv is described with phase space ℝNv × ℝNv, 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

$$H(x,\xi ) = \sum\limits_{i = 1}^N {\frac{{\xi _i^2}}{{2{m_i}}}} + \sum\limits_{i < j} {{V_{ij}}} V({x_i} - {x_j})$$
(5.0.1)

, 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

$${V_{ij}}(x) = \frac{{ - {m_i}{m_j}}}{{\left| x \right|}}$$
(5.0.2)

.

Keywords

  • Configuration Space
  • Quantum Case
  • Free Region
  • Asymptotic Completeness
  • Trapping Energy

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1997 Springer-Verlag Berlin Heidelberg

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Dereziński, J., Gérard, C. (1997). Classical N-Body Hamiltonians. In: Scattering Theory of Classical and Quantum N-Particle Systems. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03403-3_6

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  • DOI: https://doi.org/10.1007/978-3-662-03403-3_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08284-9

  • Online ISBN: 978-3-662-03403-3

  • eBook Packages: Springer Book Archive