Internal or shape coordinates in then-body problem

The construction of global shape coordinates for the n-body problem is considered. Special attention is given to the three- and four-body problems. Quantities, including candidates for coordinates, are organized according to their transformation properties under so-called democracy transformations (orthogonal transformations of Jacobi vectors). Important submanifolds of shape space are identified and their topology studied, including the manifolds upon which shapes are coplanar or collinear, and the manifolds upon which the moment of inertia tensor is degenerate.