Geodesic flow on SL(2,\({\mathbb{R}})\) with nonholonomic constraints
type: Item
instance of: scholarly article
title: Geodesic flow on SL(2,\({\mathbb{R}})\) with nonholonomic constraints
zbMATH Open document ID: 0673.58034
DOI: 10.1007/BF01247085
author: V. Gershkovich
author: A. M. Vershik
published in: Journal of Soviet Mathematics
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/188541
review text: 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.
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 58A30
Mathematics Subject Classification ID: 49K30
zbMATH DE Number: 4102140
zbMATH Keywords: left-invariant nonintegrable distribution
zbMATH Keywords: variational problem
zbMATH Keywords: with constraint
zbMATH Keywords: nonholonomic analysis
zbMATH Keywords: left-invariant metric
zbMATH Keywords: constraint
zbMATH Keywords: nonholonomic geodesic flow
MaRDI profile type: Publication
cites work: Geodesics of Pfaffians
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description: scientific article; zbMATH DE number 4102140