Editing Multipole expansion (section)

==Applications==
Multipole expansions are widely used in problems involving [[gravitational field]]s of systems of [[mass]]es, [[electric field|electric]] and [[magnetic field]]s of charge and current distributions, and the propagation of [[electromagnetic wave]]s. A classic example is the calculation of the ''exterior'' multipole moments of [[atomic nucleus|atomic nuclei]] from their interaction energies with the ''interior'' multipoles of the electronic orbitals. The multipole moments of the nuclei report on the distribution of charges within the nucleus and, thus, on the shape of the nucleus. Truncation of the multipole expansion to its first non-zero term is often useful for theoretical calculations.

Multipole expansions are also useful in numerical simulations, and form the basis of the [[fast multipole method]] of [[Leslie Greengard|Greengard]] and [[Vladimir Rokhlin (American scientist)|Rokhlin]], a general technique for efficient computation of energies and forces in systems of interacting [[particle]]s. The basic idea is to decompose the particles into groups; particles within a group interact normally (i.e., by the full potential), whereas the energies and forces between groups of particles are calculated from their multipole moments. The efficiency of the fast multipole method is generally similar to that of [[Ewald summation]], but is superior if the particles are clustered, i.e. the system has large density fluctuations.