This book reveals the French scientific contribution to the mathematical theory of nonlinear oscillations and its development. The work offers a critical examination of sources with a focus on the twentieth century, especially the period between the wars. Readers will see that, contrary to what is often written, France's role has been significant. Important contributions were made through both the work of French scholars from within diverse disciplines, and through the geographical crossroads that France provided to scientific communication at the (...) time. This study includes an examination of the period before the First World War which is vital to understanding the work of the later period. By examining literature sources such as periodicals on the topic of electricity from that era, the author has unearthed a very important text by Henri Poincaré, dating from 1908. In this work Poincaré applied the concept of limit cycle to study the stability of the oscillations of a device for radio engineering. The “discovery” of this text means that the classical perspective of the historiography of this mathematical theory must be modified. Credit was hitherto attributed to the Russian mathematician Andronov, from correspondence dating to 1929. In the newly discovered Poincaré text there appears to be a strong interaction between science and technology or, more precisely, between mathematical analysis and radio engineering. This feature is one of the main components of the process of developing the theory of nonlinear oscillations. Indeed it is a feature of many of the texts referred to in these chapters, as they trace the significant developments to which France contributed. Scholars in the fields of the history of mathematics and the history of science, and anyone with an interest in the philosophical underpinnings of science will find this a particularly engaging account of scientific discovery and scholarly communication from an era full of exciting developments. (shrink)

One of the three consequences of Einstein’s theory of general relativity was the curvature of light passing near a massive body. In 1911, he published a first value of the angle of deflection of light, then a second value in 1915, equal twice the first. In the early 1920s, when he received the Nobel Prize in Physics, a violent controversy broke out over this result. It was then disclosed that the first value he had obtained in 1911 had been calculated (...) more than a century before by a German astronomer named Johann von Soldner. The aim of this article is therefore to compare the methods used by Soldner and then by Einstein leading to this first value and to explain the importance of the doubling of this value in the framework of Einstein’s theory of general relativity. Such a consequence of this theory lies at the intersection of several scientific fields such as Mathematics, Physics, Astronomy and Philosophy. (shrink)

Frisch’s 1933 macroeconomic model for business cycles has been extensively studied. The present study is the first comprehensive mathematical analysis of Frisch’s model. It provides a detailed reconstruction of how the model was built. We demonstrate the workability of Frisch’s PPIP model without adding hypotheses or changing the value of Frisch’s parameters. We prove that (1) the propagation model oscillates; (2) the PPIP model is mathematically incomplete; (3) the latter could have been calibrated by Frisch; (4) Frisch’s analysis and demonstration (...) are based on Poincaré’s methodology. (shrink)

En 1893, « physicien-ingénieur » André Blondel invente l’oscillographe bifilaire permettant de visualiser les tensions et courants variables. À l’aide de ce puissant moyen d’investigation, il entreprend tout d’abord l’étude des phénomènes de l’arc électrique alors utilisé pour l’éclairage côtier et urbain puis de l’arc chantant employé comme émetteur d’ondes radioélectriques en T.S.F. En 1905, il met en évidence un nouveau type d’oscillations non-sinusoïdales au sein de l’arc chantant. Vingt ans plus tard, Balthasar Van der Pol reconnaitra qu’il s’agissait en (...) réalité d’oscillations de relaxation. Pour expliquer ce phénomène, il fait appel à une representation dans le plan de phase et montre que son évolution prend la forme de petits cycles. Pendant le premier conflit mondial la triode remplace peu à peu l’arc chantant dans les systèmes de transmission. Au sortir de la guerre, Blondel transpose à la triode la plupart de ses résultats par analogie avec l’arc chantant. En avril 1919, il publie un long mémoire dans lequel il introduit la terminologie « oscillations auto- entretenues » et propose d’illustrer ce concept à partir de l’exemple du vase de Tantale qui sera ensuite repris par Van der Pol et par Philippe Le Corbeiller. Il fournit alors une définition d’un système auto-entretenu assez proche de celles qui seront données par la suite par Aleksandr’ Andronov et Van der Pol. Pour étudier la stabilité des oscillations entretenues par la triode et par l’arc chantant il utilise cette fois une représentation dans le plan complexe et explicite l’amplitude en coordonnées polaires. Il justifie alors l’entretien des oscillations par l’existence de cycles qui présentent presque toutes les caractéristiques des cycles limites de Poincaré. Enfin, en novembre 1919, Blondel réalise, un an avant Van der Pol, la mise en équation des oscillations d’une triode. En mars 1926, Blondel établit l’équation différentielle caractérisant les oscillations de l’arc chantant, en tous points similaire à celle qu’obtient concomitamment Van der Pol pour la triode. Ainsi, tout au long de sa carrière, Blondel, a apporté une contribution fondamentale et relativement méconnue à l’élaboration de la théorie des oscillations non linéaires. L’objet de cet article est donc d’analyser ses principaux travaux dans ce domaine et de mesurer leur importance, voire leur influence en les replaçant dans la perspective du développement de cette théorie. (shrink)