Singularities in Classical Celestial Mechanics

(1) irii'ii = -gradiU(ql9 ..., qn), i = 1, ..., n, where gradf denotes the gradient with respect to q(. Thoughout this paper we use a single dot over a variable to represent its derivative with respect to time t and a double dot to represent its second derivative with respect to t. The potential energy U has a singularity whenever q(=q^ We write this singular set ^v = {€€(«?: *, = *,}, A = U Au. The function U is real-analytic on (R)—A. Applying the standard existence and uniqueness theorems for systems of ordinary differential equations to (1), we obtain