Classical N-Body Hamiltonians

Abstract

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

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

(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)

.