The main aim of this paper is to give some counterexamples to global invertibility of local diffeomorphisms which are interesting in mechanics. The first is a locally strictly convex function whose gradient is non-injective. The interest in this function is related to the Legendre transform. Then I show two non-injective canonical local diffeomorphisms which are rational: the first is very simple and related to the complex cube, the second is defined on the whole ℝ⁴ and is obtained from a recent important example by Pinchuk. Finally, a canonical transformation which is also a gradient (of a convex function) is provided.
@article{bwmeta1.element.bwnjournal-article-apmv72z2p153bwm,
author = {Zampieri, Gaetano},
title = {Gradients and canonical transformations},
journal = {Annales Polonici Mathematici},
volume = {72},
year = {1999},
pages = {153-158},
zbl = {0951.37017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv72z2p153bwm}
}
Zampieri, Gaetano. Gradients and canonical transformations. Annales Polonici Mathematici, Tome 72 (1999) pp. 153-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv72z2p153bwm/
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