Vector fields with distributions and invariants of ODEs
type: Item
instance of: scholarly article
title: Vector fields with distributions and invariants of ODEs
zbMATH Open document ID: 1271.34041
author: Wojciech Kryński
author: B. Jakubczyk
published in: Journal of Geometric Mechanics
publication date: 2013-06-05T00:00:00Z
review text: On a manifold \(M\) of dimension \((k+1)m+1\), a pair \((X, \mathcal{V})\) consisting of a vector field and a distribution is given. The authors call it a dynamical pair if the following conditions hold. (R1) \(\operatorname{rk}(\mathcal{V}^0=\mathcal{V})=m\), \(\operatorname{rk}(\mathcal{V}^{i+1}=\mathcal{V}^i+[X, \mathcal{V}^i])=(i+2)m\) for \(i=0,\dotsc ,m\); (R2) \(\mathcal{V}^k\oplus \operatorname{span}\{X\}=TM\). They associate to this framework a class of curvature operators and a canonical connection with respect to a canonical splitting of \(TM\). The equivalence problem, namely the (local) equivalence of two dynamical pairs, is solved by using the curvature and torsion of a principal connection corresponding to a \(\operatorname{GL}(m)\)-structure on \(M\). With this formalism there are obtained invariants of ODEs under time scale preserving equivalence, with a special emphasis on mechanical control systems and Veronese webs.
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 53A55
Mathematics Subject Classification ID: 58A30
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 34A26
zbMATH DE Number: 6170794
zbMATH Keywords: vector field
zbMATH Keywords: distribution
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: semispray
zbMATH Keywords: control system
zbMATH Keywords: G-structure
zbMATH Keywords: equivalence
zbMATH Keywords: connection
zbMATH Keywords: invariants
reviewed by: Mircea Crasmareanu
MaRDI profile type: Publication
full work available at URL: jgm.2013.5.85
OpenAlex ID: W2328908069
DOI: 10.3934/JGM.2013.5.85
description: scientific article; zbMATH DE number 6170794