The Three-Body Problem

Ours, according to Leibniz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations. Because this is particularly important for celestial mechanics, in the preliminary sections we will develop as much of the transformation theory for the Euler-Lagrange and the Hamiltonian equations as is desirable for our purposes.