Invariance and the n-body problem

Abstract

The classical theorem of E. Noether [l] on invariant variational problems states briefly that if the variational integral is invariant under an infinitesimal group of transformations, then a certain number of identities hold true. Under the additional assumption that the Euler equations for the system are satisfied, these identities reduce to expressions which are constant along the extremals. In different words, the Noether identities lead to conservation laws for the system. In this manner, E. Bessel-Hagen [2] in 1921 applied the Noether theorem using the ten parameter Galilean group to derive the ten conservation laws for the classical n-body problem involving gravitational forces. In this paper we shall consider a type of converse problem, namely how the Noether theorem can be used to deduce the general form of a Lagrangian which has specified invariance properties. In particular, we shall characterize the Lagrangians which possess the same invariance properties under the Galilean group as the Lagrangian for the n-body problem. Interestingly enough, there is a large class of Lagrangians which possess these properties. We shall show in Section 3 that these Lagrangians can be written as the difference between the classical kinetic energy and a scalar potential which depends upon the magnitudes of the relative positions, the magnitudes of the relative velocities, and scalar products between the relative positions and velocities.