Gradients and canonical transformations

Gaetano Zampieri

Annales Polonici Mathematici (1999)

  • Volume: 72, Issue: 2, page 153-158
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Zampieri, Gaetano. "Gradients and canonical transformations." Annales Polonici Mathematici 72.2 (1999): 153-158. <http://eudml.org/doc/262723>.

@article{Zampieri1999,
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 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.},
author = {Zampieri, Gaetano},
journal = {Annales Polonici Mathematici},
keywords = {non-injective local diffeomorphisms; gradients; Legendre transform; canonical transformations; Hamiltonian equations; Lagrange equation; Hessian matrix; local diffeomorphisms; global invertibility},
language = {eng},
number = {2},
pages = {153-158},
title = {Gradients and canonical transformations},
url = {http://eudml.org/doc/262723},
volume = {72},
year = {1999},
}

TY - JOUR
AU - Zampieri, Gaetano
TI - Gradients and canonical transformations
JO - Annales Polonici Mathematici
PY - 1999
VL - 72
IS - 2
SP - 153
EP - 158
AB - 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.
LA - eng
KW - non-injective local diffeomorphisms; gradients; Legendre transform; canonical transformations; Hamiltonian equations; Lagrange equation; Hessian matrix; local diffeomorphisms; global invertibility
UR - http://eudml.org/doc/262723
ER -

References

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  1. [A] V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer, 1978. 
  2. [G] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. Zbl0545.49018
  3. [GTZ] G. Gorni, H. Tutaj-Gasińska and G. Zampieri, Drużkowski matrix search and D-nilpotent automorphisms, Indag. Math. 10 (1999), 235-245. Zbl1064.14511
  4. [N] J. C. C. Nitsche, Elementary proof of Bernstein's theorem on minimal surfaces, Ann. of Math. 66 (1957), 543-544. Zbl0079.37702
  5. [P] S. Pinchuk, A counterexample to the real Jacobian conjecture, Math. Z. 217 (1994), 1-4. Zbl0874.26008
  6. [Po] A. V. Pogorelov, The Minkowski Multidimensional Problem, Wiley, 1978. 
  7. [PS] P. Pucci and J. Serrin, On the derivation of Hamilton's equations, Arch. Rational Mech. Anal. 125 (1994), 297-310. Zbl0809.70012

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