Gradients and canonical transformations

• Autorzy: Gaetano Zampieri
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

non-injective local diffeomorphisms; gradients; Legendre transform; canonical transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1999
• Tom: 72
• Numer: 2
• Strony: 153-158

Daty

wydano

1999

otrzymano

1998-09-24

Twórcy

autor

Gaetano Zampieri

• Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, I-10123 Torino, Italy

Bibliografia

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