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.
Titolo: | Gradients and canonical transformations |
Autori: | |
Data di pubblicazione: | 1999 |
Rivista: | |
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. |
Handle: | http://hdl.handle.net/11562/236600 |
Appare nelle tipologie: | 01.01 Articolo in Rivista |