J Elast (2011) 103: 247–267 DOI 10.1007/s10659-010-9283-4 Canonical and Anti-Canonical Transformations Preserving Convexity of Potentials Claude Vallée · Mohammed Hjiaj · Danielle Fortuné · Géry de Saxcé Received: 14 December 2009 / Published online: 8 December 2010 © Springer Science+Business Media B.V. 2010 Abstract The aim of the paper is to characterize transformations that preserve the poten- tial structure of a relationship between dual variables. The first step consists in deriving a geometric definition of the condition for the existence of a potential. Having at hand this formulation, it becomes clear that the canonical similitudes represents the class of trans- formations that preserves the potential form of a relationship. Next, we derive the condi- tions under which canonical similitudes preserve the convexity of the potential or change it into concavity. This new class of transformations can be viewed as a generalization of the Legendre-Fenchel transformation. These concepts are applied to the Hooke constitutive relationship. Keywords Elastic potential · Symplectic geometry · Canonical and symplectic transformations · Convexity Mathematics Subject Classification (2000) 70H15 · 26A51 · 52A41 C. Vallée Institut PPRIME, UPR 3346, SP2MI, Boulevard Pierre et Marie Curie B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France e-mail: claude.vallee@univ-poitiers.fr M. Hjiaj ( ) LGCGM/INSA de Rennes, 20 avenue des Buttes de Coësmes, 35043 Rennes cedex, France e-mail: mohammed.hjiaj@insa-rennes.fr D. Fortuné 21, rue du Hameau du Cherpe, 86280 Saint-Benoit, France e-mail: danielle.fortune123@orange.fr G. de Saxcé Laboratoire de Mécanique de Lille CNRS/UMR 8107, Boulevard Paul Langevin, 59655 Villeneuve d’Ascq Cedex, France e-mail: gery.desaxce@univ-lille1.fr