  Citation: Badur, J.; Feidt, M.; Ziólkowski, P. Duhem and Natanson: Two Mathematical Approaches to Thermodynamics. Energies 2022, 15, 1881. https://doi.org/10.3390/ en15051881 Academic Editor: Sergio Ulgiati Received: 29 December 2021 Accepted: 25 February 2022 Published: 3 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article Duhem and Natanson: Two Mathematical Approaches to Thermodynamics Janusz Badur 1 , Michel Feidt 2 and Pawel Ziólkowski 3, * 1 Energy Conversion Department, Institute of Fluid Flow Machinery, Polish Academy of Science, Fiszera 14, 80-231 Gda ´ nsk, Poland; jb@imp.gda.pl 2 Laboratory of Energetics & Theoretical & Applied Mechanics, University of Lorraine, 2 Avenue de la Forêt de Haye, 54518 Vandœuvre-lès-Nancy, France; michel.feidt@univ-lorraine.fr 3 Faculty of Mechanical Engineering and Ship Technology, Institute of Energy, Gda´ nsk University of Technology, Narutowicza 11/12, 80-233 Gda´ nsk, Poland * Correspondence: pawel.ziolkowski1@pg.edu.pl Abstract: In this article, the previously unrecognized contributions of Pierre Duhem and Ladislavus Natanson in thermodynamics are shown. The mathematical remodelling of a few of their principal ideas is taken into consideration, despite being neglected in the literature. To emphasize these ideas in an appropriate epistemological order, it would be crucial to first revalue and reconstruct some underrepresented parts of the proceedings process through which Duhem and Natanson created their thermodynamics. Duhem and Natanson’s scientific works are against the background of modern continuum mechanics, presenting relevant approaches. In line with the long-held beliefs of many French and Polish researchers, the article mentions that Duhem and Natanson’s ideas dated back to one century ago. Both scientists were qualified in the same Royal Way, which in this case includes chemistry, mechanic of fluid and solid, electro-chemistry, thermodynamics, electrodynamics, and relativistic and quantum mechanics. Therefore, it is possible to connect and then compare the results of their conceptions and approaches. Duhem and Natanson are both in firm opposition with Newtonian mechanisms. Thus, the Maupertuis least action principle created the ground for their efforts, in which they flourished as an elementary quantum. Keywords: thermo-chemistry thermodynamics of hidden parameters; Duhem inequality; Natanson’s fundamental equation; non-linear evolution equations; logical structure of the extended thermo- dynamics; Maxwell-Natanson un-objectivity; Zaremba relativity; co-rotational time derivative; objective thermodynamics; nonlocal thermodynamics of Navier-Stokes-Fourier fluid 1. Introduction Cimmelli, Jou, Rugerri, and Ván[1] recently elaborated concise versions of the modern mathematical methods used in thermodynamics. To do this, they surveyed numerous results from classical, irreversible, extended, and statistical thermodynamics in order to obtain a summary of the current methods and their usefulness as scientific tools. Here, we take a similar approach, but instead outline the state of thermodynamics at the end of the 19th century. We begin by introducing the state-of-art for the period of 1870–1880, before describing the thermodynamic models developed by Duhem and Natanson. Here, we aimed to reconstruct and revalorize two extremely abstract, potential-based thermodynamic models that were created simultaneously by two independent scientists: Pierre Duhem (Lille, Bordeaux, and Toulouse) and Ladislavus Natanson (Cracow). We specifically wanted to show the important developments that were made during the last decades of the 19th century, and restrict oversells to some selected not yet published mathe- matical ideas devoted to Duhem and Natanson [2–5]. Thus, we could rediscover an original construction of a very general approach, which started from the established unification of thermodynamics and mechanics. Sometimes we present the original results without Energies 2022, 15, 1881. https://doi.org/10.3390/en15051881 https://www.mdpi.com/journal/energies