Geodesic flow on SL(2,\({\mathbb{R}})\) with nonholonomic constraints (Q1120877)

The article deals with examples illustrating different constructions in recently developing (due, among others, to the authors) nonholonomic analysis. Considered is behaviour of geodesics on the group SL(2,\({\mathbb{R}})\) endowed with a left-invariant metric \(\rho\), under the constraint of the geodesics being tangent to a 2-dimensional, left- invariant nonintegrable (i.e., nonholonomic) distribution V in the tangent bundle (determined by a 2-dimensional subspace \(v\subset sl(2,{\mathbb{R}})\) such that \(v+[v,v]=sl(2,{\mathbb{R}})).\) Constructed is a nonholonomic geodesic flow (n.g.f.) in the mixed bundle over SL(2,\({\mathbb{R}})\), having fibre \(V\oplus V^{\perp}\) (the so-called ``centaurus''). Parallelization of this bundle yields a bundle over \(v\oplus v^{\perp}\), with fibre SL(2,\({\mathbb{R}})\); thus n.g.f. is, as dynamical system, the skew product of a flow on \(v\oplus v^{\perp}\), with the mentioned fibre. The motion in the base is qualitatively characterized using a classification of such distributions V and metrics \(\rho\), while for the motion in the fibre the same is done by constructing certain monodromy mapping and its monodromy submanifolds. Finally, a connection of n.g.f. over SL(2,\({\mathbb{R}})\) with flows on the Lobatchevsky's plane is explained.