  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.
