The application of variational principles in continuum mechanics is in the mainstream of mechanical science still highly controversial. Sometimes such principles are even rejected as being logically inconsistent and not physically well-grounded. However, many of such questions have already been answered in elder papers not written in English. A crystal-clear article on that topic is the fundamental review article of Ernst Hellinger “Die allgemeinen Ansätze der Mechanik der Kontinua”. Following his predecessors J.-L. Lagrange, G. Piola and the Cosserat brothers, Hellinger presents in this treatise a nonlinear field theory of continuum mechanics entirely based on a variational principle "the principle of virtual displacements".
The collection of forces and stresses of any kind [...] have in common, that for any virtual displacement they expend a virtual workE. Hellinger, Die allgemeinen Ansätze der Mechanik der Kontinua, 1913
This exegetic series intends
i) to allow to those who cannot understand German to enjoy the reading of a crystal-clear and still topical article whose content has some enlightening parts,
ii) to trace the origins of current ideas of mechanical sciences to their original sources,
iii) to show that only one century ago the principle of virtual work (or virtual displacements/velocities) was regarded as the central principle in continuum mechanics and that Hellinger did forecast already then the main lines of its development,
iii) to discuss some technical and conceptual aspects of the variational principles in continuum mechanics which some authors consider still controversial.
Exegetic Series appeared in ZAMM
- Introduction and Section I
Eugster, S. R. and dell'Isola, F.: “Exegesis of the introduction and sect. I from "Fundamental of the Mechanics of Continua" by E. Hellinger”, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 97(4), pp. 477-506, 2017. PDF, ZAMM
- Section II and III.A
Eugster, S. R. and dell'Isola, F.: “Exegesis of sect. II and III.A from "Fundamental of the Mechanics of Continua" by E. Hellinger”, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 98(1), pp. 31-68, 2018. PDF, ZAMM
- Section III.B
Eugster, S. R. and dell'Isola, F.: “Exegesis of sect. III.B from "Fundamental of the Mechanics of Continua" by E. Hellinger”, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 98(1), pp. 69-105, 2018. PDF, ZAMM
<<The paper by Eugster and dell'Isola introduces Hellinger's work to the international readers. Undoubtedly, this paper is an important contribution since it points to significant and still relevant developments in mechanics, more than century old, which have not been widely available due to the language of the original work. Even more importantly, the current papers goes beyond mere translation and places into a wider context the fundamental nature of Hellinger’s work and how the field of mechanics has been poorer for lack of its ready availability. The style of the current paper makes it highly readable. The concepts are presented in a way accessible to more modern students of mechanics by relating Hellinger's work to the newer concepts and notations. The authors' unique style, their accompanying commentary, their critical interpretations and explanations, and the references to modern works, many of which inadvertently follow concepts formulated by Hellinger, makes the current paper novel and of interest from theoretical as well as practical viewpoints. The paper makes a powerful and persuasive case for the centrality of the ''variational ansatz'' to mechanics. In an era of over reliance on numerical approaches without proper attention to their foundations this paper is timely and well-written, and, therefore deserves prompt publication.>> Reviewer 1, Exegesis of Introduction and Sect. I
<<This is a history of mechanics of major importance and significance. Here the authors show the primary role played by Hellinger (and his intellectual predecessor, Piola) in formulating modern mechanics. The paper is well written and comprehensively documented. It is wonderful and engaging to read. By bringing Hellinger (and Piola before him) to the wider community, the authors do a great service. In particular, they establish beyond any doubt that contemporary 'discoverers' of important notions such as the 'part-wise virtual-work principle' are in fact merely re-discoverers of ideas already advanced by Hellinger (or Piola). In fact these revelations allow modern students to visit the original sources, and there to discover the original ideas stated clearly and without fanfare, in pleasant contrast to the modern rediscoveries. Beyond these virtues, the paper argues convincingly about the effect that changes in the 'lingua franca' can have on the evolution of science. If anything, the paper is too forgiving of the modern rediscoverers in failing to mention one very typical human failing: the need for recognition, however far one is from earning it. In my view the paper simply must be published!>> Reviewer 2, Exegesis of Introduction and Sect. I
<<The paper is a major contribution to the history of Mechanics and should be published. Like the first paper of this series, this one is a masterpiece of scholarship, one that rights many historical wrongs committed by successors in their zeal to make a mark on the subject.>> Reviewer 1, Exegesis of Sect. II and III.A
<<As the title suggests, the work contains not only the translation of Hellinger's work, but also a historical perspective on his works and of the evolution of continuum mechanics in general. The reading is very interesting and surprising in terms of contents and style. The authors balance very well translation, comments and historical considerations. The manuscript is very well written and easy to read. I consider that this work deserves publication in the present form.>> Reviewer 2, Exegesis of Sect. II and III.A
<<This is the third in a series of papers by Eugster and dell'Isola that, through translation of the work by Hellinger and the accompanying commentary by the authors, will bring to a wider audience the significant and powerful developments in mechanics, more than century old, which have not been widely available due to a variety of reasons lucidly discussed by the authors. Indeed the authors utilize the translation as a vehicle to discuss the factors that often control the state of scientific progress while faithfully and carefully discussing/commenting and highlighting the centrality of "variational ansatz" in Hellinger's approach to continuum field theory. The third paper of the trilogy by Eugster and Dell’isola then completes the arguments from both the scientific perspective and the view of socio-politics of scientific endeavors. The style and content is even more persuasive the third time. The overall 'package' conceived by the authors is novel, besides being timely in this era of proliferating publications each of which proclaims unique contributions while almost pathologically giving short shrift to the preceding contributions. The resulting trilogy will not only serve pedagogical purposes for students learning variational methods but also shed light on socio-political factors related to art of publication, progress in science and the role of dominant personalities, the 'pioneers' versus the 'colonists'. This paper, therefore, as its preceding ones in the series, deserves prompt publication. The paper has been carefully edited and suffers from only a very few minor editorial issues. This paper can be accepted as is without revision. >> Reviewer 1, Exegesis of Sect. III.B
<<This is a salacious paper in which the authors maintain that a large part of the insights given in the studies Truesdell and W. Noll, "The non-linear field theories of mechanics" and Truesdell and R. Toupin, "The classical field theories" was actually available in a review article presented in 1913 by Ernst Hellinger. One of the Authors' intent is to "prove...how derivative and unoriginal some recent and contemporary mechanicians truly are." In doing so, they provide a sarcastic description, and, to be honest, rather realistic, of what the milieu of contemporary mechanicians is like, arranged into rude gangs of Lagrangians/Hamiltonians and Truesdellians, always seeming to be balanced upon the razor edge between survival and self-destruction. [...] The fact is, I - sadly - agree with a couple of statements of the paper, that are: 1. the central role of variational methods for their heuristic potential, as shown by the history of science, and also by contemporary physics ("Lagrangians stay much more stably in the physics textbooks", i.e. molecular dynamics simulations); 2. the lack of independency of many contemporary young scholars, considering "gospel" the reference text of the clan they belong to, stimulated by our times of "fast and copious publication activity."
As in the Shakespearian Montecchi vs. Capuleti saga, well described in Baz Luhrmann’s Romeo+Juliet as a conflict between gangs, everything ends up with the destruction of both sides, and of the dream of Love beyond borders, this paper probably epitomizes the decline of mechanics as a science in which it still makes sense to do Research.>> Reviewer 2, Exegesis of Sect. III.B
Further publications and links
Eugster, S.R., dell'Isola, F.: "An ignored source in the foundations of continuum physics “Die Allgemeinen Ansätze der Mechanik der Kontinua” by E. Hellinger", In Proceedings in Applied Mathematics and Mechanics, 2 pages, in press, 2017. PDF
Presentation by Simon R. Eugster in celebration of the Tullio Levi-Civita Lectures 2016 of the International Research Center for the Mathematics & Mechanics of Complex Systems, 6. October 2016, Rome, Italy.
Göttinger Digitalisierungszentrum: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Band IV-4.