Cartan meets Chaplygin

Kurt M. Ehlers, Jair Koiller

In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ``method of equivalence'' can provide the invariants of nonholonomic systems on a manifold $Q$ with kinetic lagrangians \cite{Cartan1928}. Cartan indicated which changes of the metric outside the constraint distribution $E\subset TQ$ preserve the \emph{nonholonomic connection} $D_XY=\operatorname{Proj}_E\nabla_XY$, $X,Y\in E$, where $\nabla_XY$ is the Levi-Civita connection on $Q$ and $\operatorname{Proj}_E$ is the orthogonal projection over $E$. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems \cite{Chaplygin1, Chaplygin2, KoillerArma}. We also discuss an example-a mathematical gem!-found by Oliva and Terra \cite{Oliva}. It implies that there is more freedom (thus more opportunities) using a \emph{weaker } equivalence, just to preserve the straightest paths: $D_XX=0$. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide in the following cases: i) Rank two distributions. This implies for instance that in Cartan's example of a sphere rolling on a plane without slipping or twisting, a (2,3,5) distribution, the two notions of equivalence coincide; ii) For a rank 3 or higher distribution, the corank of D in D+[D,D] must be at least 3 in order to find examples where the two notions of equivalence do not coincide. This rules out the possibility of finding examples on (3,5) distributions such as Chaplygin's marble sphere. Therefore the beautiful (3,6) example by Oliva and Terra is minimal.