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Jacques Hadamard yg

also became Maitre de Conferences de Physique at Bordeaux in 1893, and the two friends resumed their conversations at the point where they left off in 1887. Duhem’s Lille lectures on hydrodynamics, elasticity and acoustics had been published in 1891, and Hadamard had naturally read them. In the volume of Memoires de la Societe des Sciences Physiques et Naturelles de Bordeaux dedicated to the life and work of Duhem [227] Hadamard wrote: ‘Notre reunion a la Faculte des Sciences de Bordeaux me procura la rare fortune d’en completer la lecture par de precieux et continuels echanges de vues. A cette lecture, a ces echanges de vues, je dois la plus grande par tie dc mes travaux ulterieurs, presque tous consacres au Calcul des Variations, a la theorie d’Hugoniot, aux equations aux derivees partielles hyperboliques, au principe de Huyghens’. Hadamard only remained at Bordeaux until 1897 by which time all his most famous papers on classical analysis, culminating in the proof of the prime number theorem in 1896, had been published. While at Bordeaux he published several papers on dynamics, in particular the one for which he received the Prix Bordin of the Academy. In fact the prize was for work on geodesics, and the memoir was mainly concerned with geodesics which also enter explicitly or implicitly into much of the later work which stems as we have seen from his contact with Duhem. Hadamard was awarded the Prix Petit d’Ormoy in 1903 for the whole set of his mathematical work, and in 1907 he shared the Prix Vaillant with three others.

Returning to Paris as Maitre de Conferences at the Sorbonne in October 1897, he became Professeur Adjoint in February 1900; in November 1897 he also became Suppleant to Maurice Levy in the Chair of Mecanique Analy- tique et Mecanique Celeste at the College de France.

About this time Hadamard was much concerned about the Dreyfus case which began in 1894. Dreyfus’s wife was the daughter of David Hadamard, a cousin of Jacques Hadamard’s father. It is said that Hadamard confided to his friend Painleve that he had only seen Dreyfus once and that ‘sa figure ne lui avait guere plu’. Painleve gossiped and some days later people read in the newspapers ‘raeme la famille de Dreyfus ne croit pas a son innocence’. At once Hadamard involved Painleve in the action in favour of Dreyfus. The agitations involved Zola and led to a fresh trial in 1899 with extraordinary scenes after which Dreyfus was pardoned, but he was not satisfactorily cleared until much later. During the Zola trial in 1898 the idea was launched which led to the founding of the Ligue des Droits de 1’Homme to rally ‘all those who without distinction of religious belief wish for a sincere union between all Frenchmen and are convinced that every kind of arbitrary action or intolerance threatens civil disturbance and is a menace to civilization and progress’. Hadamard sat on the Central Committee of the Ligue from its foundation until less than a year before he died when Mile Jacqueline Hadamard took his place. In the 1920’s he made several contributions to the ‘Cahiers’ of the Ligue. It was never as a relative of Dreyfus that he was concerned, but for justice itself, and throughout his busy life however many honours were showered upon him he always had the preoccupations of

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Jacques Hadamard 89

first page of Frechet’s Les espaces abstraits,‘Le continu fonctionnel . . . n’offre a notre esprit aucune image simple. L’intuition geometrique ne nous apprend rien, a priori, sur son compte. Nous sommes forces de rcmedier a cette ignorance et nous ne pouvons le faire qu’analytiquement, en creant a l’usage du continu fonctionnel un chapitre de la theorie des ensembles’.

I now come to my most difficult task, the discussion of Hadamard’s contributions to the theory of the propagation of waves and to the theory of partial differential equations which are contained mainly in Legons sur la propagation des ondes [91] and Lectures on Cauchy's problem [187, 261] with some later papers on Huygcn’s principle [204, 222]. The starting point was Hugoniot’s work on the rectilinear movement of a gas, or rather Duhem’s exposition of it in his lectures and in the conversations in the years 1893- 1897 to which I referred earlier. It emerged from Hugoniot’s posthumous memoirs that in order to express the physical situation correctly it was necessary to consider the possibility of mathematical discontinuities, and that approximations by analytic or continuous functions could not give a true representation of the physics. A thorough discussion of movements other than rectilinear movements involved mathematical difficulties far too great for most physicists of that time, but these difficulties positively attracted Hadamard and the physical difficulties attracted Duhem. The waves referred to are not those of the theory of vibrations and it may help to quote Hadamard’s description of his concept given in a lecture [152]. ‘Let a perturbation be produced anywhere, like sound; it is not immediately perceived at every other point. There are then points in space which the action has not reached in any given time. Therefore the wave, in that sense a surface, separates the medium into two portions (regions): the part which is at rest, and the other which is in motion due to the initial vibration. These two portions of space are contiguous.’

In the case of shock waves the velocity is discontinuous at the surface; gas passes through the shock front, and the equation for the medium behind the shock front is non-homogeneous. In other cases the movement was said to be ‘compatible’ with that on the other side of the wave surface.

The early history of the theory of shock waves is very complicated; many of the results were obtained independently by a second or third person many years after the original discovery was published. For instance many of Hugoniot’s results had been obtained earlier, some by Riemann in 1860 who in some respects made incorrect assumptions, some by Rankine in 1869, and as late as 1910 Rayleigh felt it necessary to show that Rankine’s equations were equivalent to Hugoniot’s and said that Rankine’s work had been neglected. Hadamard mentions the work of Riemann and Christoffel, but not that of Rankine. A full appreciation of the value of this early work on waves has only come about in comparatively recent times, during the last ten or fifteen years, say, with its application to new fields. A full account of the work of Hadamard and Duhem on the general theory of waves is given by Truesdell and Toupin. Courant and Friedrichs give an account of the

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Jacques Hadamard ggG unther, P. 1965. Ein Beispiel einer nicht trivialen Huygensschen DifTerentialgleichung

m it vier U nabhangigen V ariablen. Archive for Rational Mechanics & Analysis, 18, 103-106. H ardy, G. H. & Littlewood, J . E. 1931. Some properties of fractional integrals. II. Math.

34, 403-439.H ugoniot, H . 1887/89. Sur la propagation du mouvement dans les corps et specialement

dans les gaz parfaits. J . Ecole Polytechnique I, 57, 1-97; II, 58, 1-125.L andau, E. 1909. Handbuch der Lehre von der Verteilung der Primzahlem. Lepzig: Teubner. Ju lia , G. 1962. Speech at the presentation of the gold medal on the fiftieth anniversary of the

election of M. Jacques H adam ard to the Academy. Notices et discours, Acad. Sci. Paris, 4, 730-732.

Leray, J . 1962. C auchy’s problem . In Man, science, learning and education. (Edited by S. W.H igginbotham .) Rice University semicentennial publications.

Levy, P. 1964. Jacques H adam ard (1865-1963). Lajaune et la rouge (Journal of the Societe amicale des anciens eleves de l’Ecole Poly technique), 10, 3-9.

M andelbrojt, S. 1953. The m athem atical work of Jacques H adam ard. Amer. Math. Monthly, 60, 599-604.

M andelbrojt, S. & Schwartz, L. 1965. Jacques H adam ard (1865-1963). Bull. Amer. Math. Soc. 71, 107-129.

Morse, M. & H edlund, G. A. 1938. Symbolic dynamics. Trans. Amer. Math. Soc. 60, 815-866. Petrovskii, I. G. 1946. O n some problems of the theory of partial differential equations.

Translation Series 1, Vol. 4, Differential Equations (Providence, R .I., Amer. Math. Soc. 1962), 373-414, translated from Uspechi Mat. Nauk. (N.S.) 1, No. 3-4 (13-14), 44-70.

Poincare, H. 1885. Sur les courbes definies par les equations differentialles (3e partie). J . de Math. 4e serie, 1, 167-244.

Schwartz, L. Theorie des distributions 1 (1957), 2 (1959). Paris: H erm ann.Smithies, F. 1962. Integral equations. Cam bridge M ath. T ract 49.Stieltjes, T . J . 1885. Sur une fonction uniforme. C.R. Acad. Sci. Paris, 101, 153-154. T itchm arsh, E. C. 1939. Theory of functions. 2nd Ed. Oxford Univ. Press.Truesdell, C. & Toupin, R. 1960. T he classical field theories. Handbuch der Physik. (edited

Flugge), m/1.Truesdell, C. 1961. Archive for rational mechanics and analysis, 8, 263-296. von M angoldt, H . 1898. U eber eine Anw endung der Riem annschen Formel fur die

A nzahl der Prim zahlen un ter einer gegebenen. J . reine angew. Math. 119, 65-71.

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