Kurt Ehlers, Jair Koiller, Pedro M. Rios
Nonholonomic Systems: Cartan's equivalence and Hamiltonization
Preprint series: ESI preprints

MSC:

58F99 None of the above but in this section

70F25 Nonholonomic systems

53C20 Global Riemannian geometry, including pinching, See Also {31C12, 58B20}

53C07 Special connections and metrics on vector bundles

Abstract: A {\it nonholonomic mechanical system} is given by a configuration space $Q^n$ endowed with a riemannian metric
$\langle\,,\,\rangle$, a potential $V$, and a totally nonholonomic constraint distribution ${\cal H} \subset TQ$. Passing to the Jacobi metric, we may assume that $V=0$, so the trajectories are geodesics of a (non metric) connection $\nabla_{NH}$ which mimics the Levi-Civita connection. The dynamical system can also be described in terms of an almost Poisson tensor $\{\,,\,\}_{NH}$ with non-zero Jacobinizer. The paper is divided into two parts, related by the use of {\it moving frames} in $Q$. In the first part we explore the {\it connection viewpoint}, and is devoted to the {\it local geometric invariants} obtained via Cartan's method of equivalence; as an specific example, we analyze Engel's (2-4) distribution. In the second part, we explore the {\it almost Hamiltonian description}. When a Lie symmetry group $G$ is present, the dynamics can be either {\it reduced} (in the case of internal symmetries) or {\it compressed} (transversal symmetries). Important special cases are $G$-{\it Chaplygin systems}, in which the constraints are given by a connection on a principal bundle, with total space $Q$ and base $S = Q/G$. These systems ``compress'' to the cotangent bundle $T^*S$ of the base; in favorable cases, the compressed system is hamiltonizable.
{\it Hamiltonization of a nH system with internal symmetry can also occur on a reduced stage.} Chaplygin's homogeneous sphere, a perfect ball rolling without slipping on the plane, is non hamiltonizable in $T^*SO(3)$, but it is hamiltonizable when {\it reduced} to $T^*S^2$. We conjecture that the same could be true for the general case of unequal inertia coefficients.

Keywords: non-holonomic mechanics, Cartan's equivalence method, affine connections