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 being related to the complex cube, the second is defined on the whole $R^4$ 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.

Gradients and canonical transformations

ZAMPIERI, Gaetano
1999-01-01

Abstract

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 being related to the complex cube, the second is defined on the whole $R^4$ 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.
1999
Legendre transform; Gradients; Non-injective local diffeomorphisms; Canonical transformations
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/236600
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