2018/04/09 by Montgomery, Richard
#37J50 #37N05 #70F10 #74H35 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1804.03059
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.