×

On a remark of I. Newton and the mathematics of the Second Law. (English) Zbl 0883.01017

Martinás, K. (ed.) et al., Thermodynamics. History and philosophy: facts, trends, debates. Proceedings of the conference, Veszprém, Hungary, July 23–28, 1990. Singapore: World Scientific. 449-454 (1991).
In the last decades the general thermodynamical discussion has been encumbered by a historically deeply rooted “axiomatical paradentosis” which consists in undermining (partly by use of rather sophisticated mathematical means, e.g. the exterior differential calculus) the position of the Second Law as an axiom independent of the other axioms [cf. e.g. K. Bleuler, Differential geometric methods in various domains of theoretical physics. Rep. Math. Phys. 12, 395-405 (1977; Zbl 0374.53013) and H. J. Borchers, Some remarks on the second law of thermodynamics. Rep. Math. Phys. 22, 29-48 (1985; Zbl 0611.76007), where in effect no less than the provability of the Second Law is asserted].
For detailed historical discussions of these delicate questions cf. e.g. J. Walter, Das Gesetz von Maxwell-Clausius in the phänomenologischen Thermodynamik. Proc. Symposium, Dundee/ Scotland 1982 (W. N. Everitt and R. T. Lewis, eds.), Lect. Notes Math. 1032, 509-521 (1983; Zbl 0556.35111); On H. Buchdahl’s project of a thermodynamics without empirical temperature as a primitive concept. J. Phys. A, Math. Gen. 22, 341-342 (1989); On the mathematics of the First Law of thermodynamics. 150 years of J. R. Mayer’s “Remarks on the forces of inorganic nature” (German, translation now available), Res. Math. 28, 15-32 (1995; Zbl 0854.01014) and quite recently M. Monleón Pradas and P. Pedregal, The fundamental property of a perfect gas and the second law of thermodynamics, Arch. Ration. Mech. Anal. 136, No. 4, 383-408 (1996; Zbl 0882.76075); E. H. Lieb and J. Yngvason, The physics and mathematics of the second law of thermodynamics. Preprint No. 469 of the Erwin Schrödinger Int. Inst. Math. Phys. (Vienna, 1997).
In the present historico-mathematical essay this discussion is continued by drawing attention to I. Newton’s dictum “The resolution of the problem when the equation comprises three or more fluxions of quantities is briefly disposed of. Specifically, any relation you wish (when the conditions of the inquiry do not determine one) should be presupposed between any two of those quantities…” [cf. p. 112/113 in D. T. Whiteside (ed.): The mathematical papers of I. Newton, Vol. III. The University Press, Cambridge (1969; Zbl 0172.28301)] to which I was led by some historical remarks by A. P. Yushkevich on p. 420, 423 of V. V. Stepanov, Lehrbuch der Differentialgleichungen. VEB Deutscher Verlag der Wissenschaften, Berlin (1956; Zbl 0070.30603).
In modern language Newton’s dictum says that, in order to reduce the abundance of solution curves of the underdetermined differential equation \(v_1\cdot\dot x_1+ v_2\cdot\dot x_2+ v_3\cdot\dot x_3=0\) one should restrict consideration to the solution curves (“quantities”) lying on some arbitrarily chosen surface \(\mathcal S\). Newton himself seems to expect a hint concerning this choice to originate from the domain of (physical) applications. From my studies in the mathematics of the Second Law it turned out that it would be highly relevant to choose \(\mathcal S\) to be a thin tube. In this case the solution curves generically take on the shape of flat screws for whose pitch (“Ganghöhe”) then an elegant formula can be derived which is of use in proving the existence of the so-called “absolute temperature”.
For the entire collection see [Zbl 0879.00073].
Reviewer: J.Walter (Aachen)

MSC:

01A45 History of mathematics in the 17th century
80-03 History of classical thermodynamics

Biographic References:

Newton, I.
PDFBibTeX XMLCite