Abstract
Nonlinear oscillations appeared to be a major technological and scientific problem at the beginning of the 20th century. Until the 1920s, analysis of these oscillations remained merely empirical and technologically driven. The first attempts to theoretically encompass these technological problems led to a shared mathematical analysis of nonlinear oscillations, borrowing from the mathematics of differential equations, and putting forward concepts such as “relaxation oscillations”. We focus on the work of Nicolai Fyodorovich Minorsky (1885–1970), a Russian born naval engineer. His seminal work of 1922, «Directional stability of automatically steered bodies» illustrates how he dealt with mathematical knowledge for the steering of ships and the controlling of oscillations. We show how he progressively tackled nonlinear oscillations problems in this context. He later became an advocate of nonlinear oscillations theories, among a community that grew progressively in the 1920s. This community, although very heterogeneous, can be characterised by a strong appeal to mathematics. Minorsky’s commitment to nonlinear theories sheds light on the mathematics and epistemology of mathematics that were at stake in the community.
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Notes
- 1.
The life and work of Minorsky were earlier described in (Petitgirard 2015). We make use of this previous assessment and draw on investigations conducted with new archival materials. Minorsky’s scientific and technological output is impressive, and includes textbooks, numerous papers in very diverse academic publications and many patents.
- 2.
For the first part of Minorsky’s life, we mainly rely upon information from (Stuart 1984, pp. 10–11), (Flügge-Lotz 1971) and the recent article (Matveyev 2019). The latter was written with access to new archival materials related to Minorsky’s early years in the military domain and his inventions (archives from the Russian State Archive of Military Museums [URL: https://rgavmf.ru]). We are still looking for archives that remained to be examined from the period before 1918 and his departure from Russia. We must also emphasize that the town of his birth, Korcheva, was submerged with the building of the canal between the Volga and Moskova Rivers in 1937.
- 3.
“Junker” (or “Yunker”) has different meanings in the Russian Empire. One corresponds to the rank given to a volunteer for military service in the Imperial Russian Army. Another is a term referring to a student at a Junker School (which began to be referred to as a Military school after 1911) (Mayzel 1975). The second meaning fits the curriculum of Minorsky (Matveyev 2019).
- 4.
See Gouzévitch and Gouzévitch (2006) to get an idea of the importance of the training offered by the “Institut électrotechnique de Nancy” to students coming from East Europe and the USSR during the period 1900–1939.
- 5.
- 6.
In 1898, Krylov received a prize from the Royal Institution of Naval Architect, for his theoretical work on shipbuilding, which extended the investigations made by William Froude. Krylov had taught at the Naval Academy in Saint-Petersburg. Even if we cannot state for sure that Minorsky followed his courses, it is very highly probable.
- 7.
- 8.
The material later used for the booklet was first published in Notes on Hydrography (Записки по гидрографии), issue 40, volume 2. (Minorsky 1916).
- 9.
The gyroscope was built to determine an angular position, while the gyrometer gives an angular velocity. (Matveyev 2019) gives many details about Minorsky’s early inventions.
- 10.
It corresponds to the USA patents n° 1,255,480 and 1,279,411. Elmer Sperry was a prolific engineer; see (Hughes 1971).
- 11.
The US Patent is n° 1,306,552. Minorsky still declares a postal address in Petrograd in the patent: it was prepared before June 1918 and his arrival in the USA.
- 12.
Minorsky arrived in New York in June 1918, along with his spouse and their son, with the help of his brother Vladimir Fedorovich Minorsky: the latter was a famous Orientalist, an expert on Persian history and culture, and was working at the Russian embassy in Teheran (Persia) at that time.
- 13.
We refer to (Bennett 1979, 1993) in covering the period 1800–1955, and to his paper specifically addressing Minorsky’s work (Bennett 1984). The major investigations by J.C. Maxwell (1831–1879) are published in his paper “On governors” of 1868 (Maxwell 1868). Maxwell introduced several linear differential equations of control systems: at that time, stability was determined with the roots of the characteristic equation and the system would turn unstable when the real part of a complex root became positive. The British mathematician Edward Routh (1831–1907) took on this work in his Treatise on the stability in 1877 (Routh 1877). The German mathematician Adolf Hurwitz (1859–1919) independently came up with the same criteria in 1895 (hence the name “Routh-Hurwitz criteria”), (Hurwitz 1895).
- 14.
“Survival at sea pits man against nature more than it does man against man. It is not surprising that naval officers would do pioneering work in navigation, meteorology, and engineering and that the Navy would build enduring scientific institutions such as the Naval Observatory and the Naval Research Laboratory. […] Moreover, the construction and maintenance of ships require the preservation within the Navy of engineering skills firmly based on scientific disciplines.” (Sapolsky 1990, p. 5).
- 15.
- 16.
In this case, Eq. (1) gives us \( A\frac{{{\text{d}}^{2} \alpha }}{{{\text{d}}t^{2} }} + B\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} + km\alpha = D \). The characteristic equation of the differential equation is thus \( Ar^{2} + Br + kmr = 0 \), which has two complex roots \( r_{1,2} = \gamma \pm i \cdot \mu \). As the general solution of (1) takes the form \( \alpha (t) = e^{(\gamma \pm i,\mu )t} \), the real part of those roots (γ) corresponds to the damping factor. In this case, \( = \frac{ - B}{2A} \).
- 17.
(Minorsky 1922, p. 300): “[…] from which follows the remarkable result that such a disturbance has no influence upon the performance of the device, depending solely upon the inertia A of the ship, the resistance B and the constants, m, n, p, representing the intensities of the corresponding components of the control.”
- 18.
“A complete solution of the auxiliary equation of the third degree is not necessary […] M. A. Blondel has shown that by applying the so-called Hurwitz theorem of Analysis, the “stability of roots” of an algebraic equation of the nth degree can easily be established.”. Here, Minorsky cites (Blondel 1919). The Hurwitz theorem refers to the conditions of stability established by the mathematician Adolf Hurwitz in 1895 (in his paper (Hurwitz 1895)). When the characteristic equation of a differential equation, with real coefficients, is expressed: \( a_{n} r^{n} + a_{n - 1} r^{n - 1} + \cdots + a_{1} r + a_{0} = 0, \) the Hurwitz condition is expressed with relations between the an. The first is: all the an > 0; the second is expressed by combinations of an. When the equation is of order 3, the conditions are simply: an > 0 and (a1.a2 - a0 a3) > 0. In the “class 2” case, the conditions are: B + kp > 0; (B + kp).kn − Akm > 0; km > 0. (Minorsky 1922, pp. 301–302).
- 19.
- 20.
(Minorsky 1922, p. 283). Minorsky emphasizes: “for in the case of unlimited angular motion […] there is no analytical expression applicable to the various torques acting on a ship in general.”
- 21.
The tests are detailed in (Minorsky 1930). They were based on two US patents: n° 1,436, 280 «Automatic steering device» (this was the first definition of an automatic system, going back to previous results of 1918 and benefiting from the analysis published in 1922—the patent was granted in November 1922); and patent n° 1, 703,280 “Directional stabilizer” issued in September 1922.
- 22.
For details, see (Ginoux and Petitgirard 2010).
- 23.
For example a multivibrator oscillates between two states. Oscillating devices with two stable equilibrium states are said to be bistable. Discontinous oscillations refers to an oscillating system with slow variations but showing «jumps» or very rapid changes at a certain time.
- 24.
- 25.
Regarding the «arc hysteresis» phenomenon, see (Ginoux 2017, p. 18) and the following pages.
- 26.
Janet decided to model the nonlinear characteristic of the triode through a series of even terms that “will certainly converge.”
- 27.
In (Van der Pol 1920), van der Pol chose to approximate the characteristic equation of the triode by a third order Taylor-MacLaurin expansion, relating i (intensity) and v (voltage) by the formula: \( i = - \alpha v + \beta v^{2} + \gamma v^{3} \). The differential equation of the triode in the circuit was thus:
\( C\frac{{{\text{d}}^{2} v}}{{{\text{d}}t}} + \left( {\frac{1}{R} - \alpha } \right)\frac{{{\text{d}}v}}{{{\text{d}}t}} + \frac{1}{L}v + \beta \frac{{{\text{d}}(v^{2} )}}{{{\text{d}}t}} + \gamma \frac{{{\text{d}}(v^{3} )}}{{{\text{d}}t}} = 0. \)
- 28.
Ginoux gives a complete analysis of the proofs of the Cartans, of van der Pol’s graphical integration and of Alfred Liénard’s (1928) results regarding the existence and uniqueness of the stable periodic solution. See (Ginoux 2017)—Chapter 3.
- 29.
Andronov delivered his analysis in August 1928 at the physicists’ congress in Moscow. It was transmitted and translated into French the year after (note to the French Academy of Science, 1929, entitled “Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues” (Andronov 1929). See (Ginoux 2017, pp. 132–136).
- 30.
We have mentioned some experts whose work is considered as a reference (even in the Soviet world): Balthasar van der Pol (1889–1959), Alfred Liénard (1869–1958), and Philippe Le Corbeiller (1891–1980). This list is not exhaustive. It should at least include other French scientists: Jules Haag (1882–1953) for his work in mechanics; the physicist Ernest Esclangon (1876–1954) for his work on “quasi-periodic” solutions; and the mathematician (and astronomer) Pierre Fatou (1878–1929). (Ginoux 2017) provides an extensive gallery of portraits.
- 31.
See (Ginoux 2012). The conference was held at the Institut Henri Poincaré (Paris) and organized by Van der Pol and the Russian mathematician Nikolaï Papaleksi. Among the participants were Alfred Liénard, Élie and Henri Cartan, Henri Abraham, Eugène Bloch, Léon Brillouin, and Yves Rocard. This event highlights the role of the French scientific community in the development of the theory of nonlinear oscillations.
- 32.
- 33.
NRL was the first laboratory created inside of a department of the US Navy, in 1923.
- 34.
DTMB was a test basin for ships of the US Navy (today it is the Carderock Division, Naval Surface Warfare Center). See (Carlisle 1998).
- 35.
We do not give details on these calculating devices in this paper, because Minorsky only developed them from 1934 until 1947. See (Petitgirard 2015). Minorsky designed a calculating device that he called a “Dynamical analogue” while he was staying at MIT’s Electrical Engineering Department directed by Vannevar Bush, in the period 1934–35. Minorsky was inspired by Bush’s then-very famous “differential analyzer”, a landmark in the history of analogue computing.
- 36.
His contribution to physics was only minor. His publications are listed in (Flügge-Lotz 1971). Minorsky tackled questions from electromagnetism, electronics, automation, etc., many of which straddle the two domains of physics and engineering sciences.
- 37.
It is worth mentioning that Minorsky married a French woman, Madelaine Palisse, in his second marital union (after Helen Ungern Minorsky left to live in Paris in 1923).
- 38.
It was first edited as a “restricted report” by the DTMB, US Navy in 1944, then republished in 1947 for a wider audience (Minorsky 1947).
- 39.
This model for rolling is a transposition of the «Froude pendulum» named after the British naval architect William Froude (1810–1879). The Froude pendulum is a frictional mechanical system that is «auto-oscillating» . The first analysis in terms of auto-oscillations is credited to (Strelekov 1933).
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Petitgirard, L. (2021). The Mathematics of Nonlinear Oscillations in the 1920s: A Decade of Trials and Convergence? Examples of the Work of Nicolai Minorsky. In: Mazliak, L., Tazzioli, R. (eds) Mathematical Communities in the Reconstruction After the Great War 1918–1928. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-61683-0_8
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