Vector fields with distributions      and invariants of ODEs

BronisŁaw Jakubczyk; Wojciech Kryński

doi:10.3934/jgm.2013.5.85

Article Contents

2013, Volume 5, Issue 1: 85-129. Doi: 10.3934/jgm.2013.5.85

This issue Previous Article Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups Next Article Computing metamorphoses between discrete measures

Vector fields with distributions and invariants of ODEs

BronisŁaw Jakubczyk^1, and
Wojciech Kryński^1,

1.
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa

Received: June 30, 2012

Revised: January 31, 2013

Published: February 2013

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Abstract

We study dynamic pairs $(X,V)$ where $X$ is a vector field on a smooth manifold $M$ and $V\subset TM$ is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. In particular, we assign to $(X,V)$ a canonical connection and a canonical frame on a certain frame bundle. We compute the curvature and torsion. The results are applied to the problem of time scale preserving equivalence of ordinary differential equations and of Veronese webs. The framework of dynamic pairs $(X,V)$ is shown to include sprays, control-affine systems, mechanical control systems, Veronese webs and other structures.

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Mathematics Subject Classification: Primary: 34C14, 53A55; Secondary: 58A30.

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