Metric cones, N-body collisions, and Marchal's lemma

Marchal's lemma is the basic tool for eliminating collisions when using the direct method of the calculus of variations to establish existence of "designer" solutions to the classical N-body problem. Our goal here is to understand why Marchal's lemma holds, by taking a metric geometry perspective and employing the Jacobi-Maupertuis [JM] metric reformulation of mechanics. Using analysis inspired by the conical metric nature of the standard Kepler problem at zero-energy, we are able to manufacture potentials, or "counterexamples", for which Marchal's lemma fails. These counterexamples overlap significantly with results obtained by Barutello et al. A novel feature in our proof for the counterexample is the use of piecewise constant potentials, and the resulting piecewise constant metrics.