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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.