Everyday physics of extended bodies or why functionals need analyzing

Document Type : Research Paper


Professor Emeritus, Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.


 Functionals were discovered and used by Volterra over a century ago in his study of the motions of viscous elastic materials and electromagnetic fields. The need to precisely account for the qualitative effects of the cohesion and shape of the domains of these functionals was the major impetus to the development of the branch of mathematics known as topology, and today large numbers of mathematicians still devote their work to a detailed technical analysis of functionals. Yet the concept needs to be understood by all people who want to fully participate in 21st century society. Through some explicit use of mathematical categories and their transformations, functionals can be  treated in a way which is non-technical and yet permits considerable reliable development of thought. We show how a deformable body such as a storm  cloud can be viewed as a kind of space in its own right, as can an interval of time such as an afternoon; the infinite-dimensional spaces of configurations of the body and of its states of motion are constructed, and the role of the infinitesimal law of its motion revealed. We take nilpotent infinitesimals as given, and follow Euler in defining real numbers as ratios of infinitesimals.


[1] Bell, J.L., A Primer of Infinitesimal Analysis", Cambridge University Press, Cambridge,1998.
[2] Kock, A., Synthetic Differential Geometry", Cambridge University Press (2nd Edition),Cambridge, 2006.
[3] Lavendhomme, R., Basic Concepts of Synthetic Differential Geometry", KluwerAcademic Publishers Dordrecht, Kluwer Texts in Mathematical Sciences 13, 1996.
[4] Lawvere, F.W., Toward the description in a smooth topos of the dynamically possiblemotions and deformations of a continuous body, Cah. Topol. Geom. Differ. Categ.XXI (1980), 337-392.
[5] Lawvere, F.W., Comments on the development of topos theory, in the book:Development of Mathematics 1950-2000", J.P. Pier (Ed) Birkhauser Verlag, Basel,(2000), 715-734. See also in TAC Reprints 24 (2014), 1-22 (with author commentary).
[6] Moerdijk, I. and Reyes, G., Models for Smooth Infinitesimal Analysis", Springer-Verlag, New York, 1991.