# Gradients and canonical transformations

Annales Polonici Mathematici (1999)

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

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topZampieri, 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

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