# American Institute of Mathematical Sciences

March  2013, 5(1): 85-129. doi: 10.3934/jgm.2013.5.85

## Vector fields with distributions and invariants of ODEs

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

Received  July 2012 Revised  February 2013 Published  April 2013

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.
Citation: BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85
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