Jacobi's criticism of Lagrange: The changing role of mathematics in the foundations of classical mechanics (Q1273193)

In the first two parts the author discusses Lagrange's understandig of analytical mechanics and shows that this understanding leads to serious foundational problems. He then sketches how Jacobi's attitude towards mathematical physics changes. The fourth and fifth part present the core of Jabobi's criticism, i.e. the analysis of Lagrange's so-called ``demonstrations'' of his priciple. The sixth chapter, finally, argues that Jacobi's criticism is based on his changing attitude towards the role of mathematics in mechanics, and the author outlines its philosophical relevance. During the centuries, scientists collected an enormous amount of insights. Finally, these would result impractical, even useless, if at the same time they would not have succeeded in combining at least most of them in a kind of ``over-theory'', in a ``principle''. This procedure sometimes failed, of course, but in total it refined scientific methods to have a far reaching effect on their neighbouring areas. Here, Lagrange's principle still plays a central role. Its overwhelming success, however, does of course not justify any conclusion on a ``world formula''. And here Jacobi's critcism comes in: Lagrange's demonstrations prove -- nothing. If Lagrange believes that he calculates nature we must today honestly admit that we can only calculate what we see from nature. Being aware of this fact we even proceed abstracting our subjective image until it is just able to answer the one question we actually have. We then call this a ``model'' where the axioms (the unprovable basic theorems) are applied to and which includes the ``force laws'' as well as, if need be, ``constraints'', both criticized in Lagrange's textbook by Jacobi. It is, of course, evident that mathematics cannot reveal modeling aspects, and Lagrange did not (could not) draw a clear dividing line between model and axiom. However, he was obviously aware of many of the basic questions: ``The only difficulty is then to find the analytical expression of those forces which are assumed to act on the bodies'', he said, and continued that ``one has then to insert the minimum number of undetermined variables'' (1764) which clearly refers to force modeling aspects and to implicitly constrained systems (the latter has not at all later been invented by Jacobi). Lagrange reduces himself to analytical force laws -- is it allowed to use this against him? Jacobi furthermore rejects Lagrange's explicit constraints \(u=0\) because these do, according to Jacobi, not tell the whole story if the body under consideration loses contact. However, Lagrange only discusses the case ``where a body is forced to move on a given surface'' (1815,TII,I,4). Jacobi's criticism is, therefore, not fair and his own discussion, on the other hand, is not complete (\(u\geq 0\Rightarrow\lambda \geq 0 \wedge u \cdot \lambda = 0\) is missing). But did he really want to bring Lagrange to fall? This question is not easy to answer. We have here two personalities with very different character: Lagrange who is told a kind person, restrained, sometimes melancholy, and Jacobi, who did not succeed to win much friends during life due to his rough and sometimes unpolite nature. And one has to have in mind that Jacobi's lectures, the basis of the present discussion, are printed from a word-for-word handwriting of one of his students (edited by H. Pulte himself, with plenty of historical footnotes -- an enjoyable reading!). Lagrange's attempts to base his considerations on calculus and to reduce the latter to algebra can only be understood by his intention to avoid ``infinitesimals'' by means of series expansions (he failed, as well known). But for the same reason Poinsot, who aimed at a ``better foundation of Lagrange's principle'', replaced Lagrange's virtual velocity ``which leaves something obscure in one's mind'' (1837, Note II) by the actual velocity -- this is an unforgiveable mistake! Finally, Jacobi's objection that ``the constitution of bodies (whether their elements are inflexible ...) is merely replaced by the defined equation of constraint'' represents a ``modern viewpoint'' of some physicists but not at all that one of Lagrange, who e.g. calculates the motion of (inflexible) gyro in a direct way both with the use of minimal coordinates and quasi-coordinates and evaluates the equations ``which coincide with those used by Euler'' (1815,TII,IX). One cannot but state Lagrange's fairness and his historical overview is, in this sense, much more than a partial substitute of missing (philosophical) preliminaries. His effort is, as pointed out in his outline, the search for a general theory ``which frees us from the burden of particularities'' (E. Becker), but by no means did he try to deduce dynamics from statics. Dismissing Lagrange's Analytical Mechanics as an ``overambitioned exercise'', his ``Euclidean method'' as ``puritanic and antispeculative'', to speak about ``rubber Euclideanism'' in the sense of stretching the borders of self-evidence (these are citations, not Jacobi's nor the author's own words--), all this reminds at modernisms like, for instance, 130 years later dismissing E. Schrödinger's statement ``that we are just beginning to collect reliable material in order to combine our total knowledge'' by the thoughtless interpretation of being a ``search for a reductional unified science with normalizing tendency in a world of such wonderful variety'' (S. Gould). Mathematics has to be unfailible -- physics cannot. Jacobi developed a critical attitude to the problem why mathematics, as a result of human thinking, should be applicable to nature. However, he pretty mixes up modeling and axioms and the applicability to both. The ``crisis of classical mechanics'' was nothing but a shock in the blind belief on the validity of Newton-Eulerian axioms in nature as it obviously also happened to Jacobi. But we have to pose the question why he needed Lagrange to put his discomfort into words. Did his disappointment about, in this sense, the helplessness of mechanics lead him to sacrifice Lagrange? Maupertuis still wanted to prove God's existence -- Lagrange did not at all. Helmut Pulte's excellent paper opens a discussion where not only historians and philosophers should take part but also mathematicians and representatives of mechanics.