Variational principles are a powerful tool also for formulating field theories
Variational principles and calculus of variations have always been an important tool for formulating mathematical models for physical phenomena. Variational methods give an efficient and elegant way to formulate and solve mathematical problems that are of
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and Luca Placidi
†‡
*
University of Roma “La Sapienza” International Telematic University “Uninettuno” International Research Centre on “Mathematics & Mechanics of Complex Systems” M& MOCS †
‡
Abstract Variational principles and calculus of variations have always been an important tool for formulating mathematical models for physical phenomena. Variational methods give an efficient and elegant way to formulate and solve mathematical problems that are of interest for scientists and engineers and are the main tool for the axiomatization of physical theories.
1 1.1
Introduction and historical background Metrodoron and his followers
The ideas we want to evocate in this lecture are very old and were put forward already in the hellenistic period: for a detailed discussion about this point the reader is referred to the beautiful book by Lucio Russo (2003). In that book it is established that “modern” science actually was born in the hellenistic era, when Metrodoron lived. Metrodoron was a pupil of a famous greek philosopher, Epicurus, and, in our opinion, the following Metrodoron’s sentence is a statement (the first?) belonging to the modern philosophy of science:
“Always remember that you were born mortal and such is your nature and you were given a limited time: but by means of your reasonings about Nature you could rise to infinity and to eternity
F. dell'Isola et al. (eds.), Variational Models and Methods in Solid and Fluid Mechanics © CISM, Udine 2011
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F. dell’Isola and L. Placidi and you indeed contemplate “the things that were, and that were to be, and that had been before””. Metrodoron Gnomologium Epicureum Vaticanum X (fr.37 Alfred K¨ orte, Metrodori Epicurei Fragmenta, “ Jahrb¨ ucher f¨ ur classiche Philologie”, Suppl. XVII, 1890, p. 557).
This dictum, following K¨ orte, comes from a lost letter or book by Metrodoron (the Epicurean philosopher) addressed to Menestratos who was presumably one of his pupils. The words quoted in bold are a citation from Iliad, I 70 (the translation into English of the sentence in boldface is ours; except for this citation the translation has been taken from Homer by Murray (1924), see the ref. (14)). In different words, Metrodoron states that by using (the right!) equations you can forecast future behavior of physical systems. 1.2
Why Variational Principles and Calculus of Variation?
In recent time, a lost Archimedes’ book (19) has been rediscovered. Some authors claim that Archimedes seems to have solved, in this book and using a variational principle, the technological problem of finding the optimal shape of a boat. Archimedes seems to have chosen, as optimality criterion, that the vertical position must be a “very” stable configuration (see Rorres (2004)). In the book of Russo (21) it is demonstrated in even a more convincing way that many optimization techniques were well-known in hellenistic science. In particular Russo proves that the problem of the determination of the regular polygon having maximal area has been solved in that period. Thus, the use of
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