Gradients and canonical transformations

Volume 72 / 1999

Gaetano Zampieri Annales Polonici Mathematici 72 (1999), 153-158 DOI: 10.4064/ap-72-2-153-158


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.


  • Gaetano Zampieri

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