Geodesic flow on SL(2,\({\mathbb{R}})\) with nonholonomic constraints. (English. Russian original) Zbl 0673.58034

Vershik, A. M.; Gershkovich, V. Ya.

J. Sov. Math. 41, No. 2, 891-898 (1988); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 7-17 (1986).

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.

Reviewer: P.Mormul

MSC:

37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25	Geodesic flows in symplectic geometry and contact geometry
58A30	Vector distributions (subbundles of the tangent bundles)
49K30	Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)

Keywords:

left-invariant nonintegrable distribution; variational problem; with constraint; nonholonomic analysis; left-invariant metric; constraint; nonholonomic geodesic flow

References:

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