Geodesic
Gradients and canonical transformations
Annales Polonici Mathematici, Tome 72 (1999) no. 2, pp. 153-158

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/ap-72-2-153-158
Mots-clés : non-injective local diffeomorphisms, gradients, Legendre transform, canonical transformations 
Gaetano Zampieri. Gradients and canonical transformations. Annales Polonici Mathematici, Tome 72 (1999) no. 2, pp. 153-158. doi: 10.4064/ap-72-2-153-158
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