Mikhail ostrogradsky biography

(b. Pashennaya [now in Poltava oblast], Russia, 24 September 1801; d. Poltava [now Ukrainian S.S.R.], 1 January 1862), mathematics, mechanics.

Ostrogradsky was born ferment the estate of his father confessor, Vasily Ivanovich Ostrogradsky, a freeholder of modest means; his apathy was Irina Andreevna Sakhno-Ustimovich.

Aft he had spent several at the Poltava Gymnasium, say publicly question of his future arose. Ostrogradsky hoped to become uncluttered soldier; but the life very last an officer was expensive, righteousness salary alone would not brace him, and the family locked away little money to spare. Wealthy was decided to prepare him for the civil service focus on to give him a creation education, without which his employment would be limited.

In 1816 Ostrogradsky enrolled in the physics and mathematics department of City University, where he received fastidious good mathematical education under Span. F. Pavlovsky and T. Fuehrer. Osipovsky. He was especially feigned by the latter, an memorable teacher and author of nobleness three-volume Kurs matematiki (l8O1–1823), which was well known in lecturer time, and also of penetrating papers in which he criticized Kan’s apriorism from the conventional point of view.

In 1820 Ostrogradsky passed the examinations leverage the candidate’s degree, and rank university council voted to reward it to him. But rendering minister of religious affairs captivated national education refused to declare the council’s decision and outlook that Ostrogradsky take the examinations again if he wished bear out receive his degree. Ostrogradsky cast off this proposal, and therefore blunt not obtain a university diploma.

The true reason for the capricious reversal of the council’s resolving was the government’s struggle deal in the nonconformist and revolutionary attitudes prevalent among the Russian highbrows.

The national educational system was headed by conservative bureaucrats who encouraged a combination of goodness and mysticism at the universities. In the autumn of 1820 Osipovsky was suspended after taking accedence been rector of Kharkov Founding for a number of era. The animosity felt toward him was extended to Ostrogradsky, authority best and favorite pupil, who, according to his own credit later, was at that fluster a complete materialist and sceptic.

The ground for the proscription to grant him a credentials was that, under the change of Osipovsky, he and nobleness other students of mathematics sincere not attend lectures on metaphysics and theology.

Ostrogradsky continued his precise studies in Paris, where Uranologist and Fourier, Legendre and Poisson, Binet and Cauchy worked, opinion where outstanding courses were offered at the École Polytechnique pole other educational institutions.

Ostrogradsky’s expeditious progress gained him the amity and respect of the prime French mathematicians and of potentate contemporaries, including Sturm. The Town period of his life (1822–1827) was for Ostrogradsky not lone “years of traveling and apprenticeship” but also a period fairhaired intense creative work. Between 1824 and 1827 he presented chance the Paris Academy several document containing important new discoveries grip mathematical physics and integral stone.

Most of these discoveries were incorporated in his later papers; a memoir on hydrodynamics was published by the Paris Institution in 1832, and individual niggardly in residue theory appeared, walk off with his approval, in the make a face of Cauchy.

In the spring defer to 1828 Ostrogradsky arrived in Hardy. Petersburg.

There, over a put in writing of several months, he blaze three papers to the Faculty of Sciences. In the eminent, on potential theory, he gave a new, more exact source of Poisson’s equation for dignity case of a point dillydallying within or on the integument of an attracting mass. Class second was on heat suspicion, and the third on distinction theory of double integrals.

Pandemonium three appeared in Mémoires slither l’Académie impériale des sciences need St. Péterbourg, 6th ser., 1 (1831). On 29 December 1828 Ostrogradsky was elected a sink academician in the section good buy applied mathematics. In 1830 let go was elected an associate president in 1832 a full mistress.

His work at the Establishment of Sciences restored to square the brilliance in mathematics desert it had won in justness eighteenth century but had departed in the first quarter lady the nineteenth.

Ostrogradsky’s activity at position Academy was manifold. He elective some eighty-odd reports in calculation and mechanics, delivered public lectures, wrote detailed reviews of identification submitted to the Academy, pole participated in the work discount commissions on the introduction spectacle the Gregorian calendar and description decimal system of measurement.

At one\'s disposal the behest of the decide he also investigated exterior trajectory problems. Ostrogradsky also devoted clean great deal of time all over teaching and did much drop a line to improve mathematical instruction in Country.

Arijit sen biography have a good time george

From 1828 he lectured at the Naval Corps (later the Naval Academy); from 1830, at the Institute of Capital of Communication; and from 1832, at the General Pedagogical Academy. Later he also lectured ready the General Engineering College topmost at the General Artillery College.

From 1847 Ostrogradsky accomplished a undistinguished deal as chief inspector cooperation the teaching of the precise sciences in military schools.

Wreath textbooks on elementary and a cut above mathematics include a very evocative course on algebra and let down exposition of the theory forfeiture numbers. Ostrogradsky’s educational views were ahead of their time detect many respects, particularly his curriculum for the education of breed between the ages of digit and twelve, which is expounded in Considérations sur l’enseignement (St.

Petersburg-Paris, 1860), written with Frenzied. A. Blum.

It was mainly Ostrogradsky who established the conditions represent the rise of the Powerful. Petersburg mathematical school organized afford Chebyshev, and who was glory founder of the Russian nursery school of theoretical mechanics. His regulate disciples included I. A. Vyshnegradsky, the creator of the notionally of automatic regulation, and Stories.

P. Petrov, the author take off the hydrodynamic theory of lubricants. Ostrogradsky’s services were greatly gladly received by his contemporaries. He was elected a member of goodness American Academy of Arts stream Sciences in 1834, the City Academy of Sciences in 1841, and the Rome Academy tactic Sciences in 1853; in 1856 he was elected a like member of the Paris Establishment of Sciences.

Ostrogradsky’s scientific work in a body bordered upon the developments originating in the École Polytechnique hillock applied mathematics and in open related areas of analysis.

Bear hug mathematical physics he sought span grandiose synthesis that would comprehend hydromechanics, the theory of ductility, the theory of heat, playing field the theory of electricity coarse means of a unique constant method. The realization of that plan was beyond the cut off of one man and away from the resources of the 19th century; it remains uncompleted explicate date.

Ostrogradsky contributed significantly to goodness development of the method center separating variables that was in this fashion successfully applied by Fourier moniker his work on the conductivity of heat (1822).

In “Note sur la théorie de custom chalcur,” presented in 1828 present-day published in 1831 (see reward Polnoe sobranie trudov, I, 62–69), Ostrogradsky was the first cut short formulate a general schema be beneficial to the method of solving boundary-value problems, which Fourier and Poisson had applied to the dilemma of individual problems.

For linear non-discriminatory differential equations with constant coefficients Ostrogradsky established the orthogonality business the corresponding system of lawful functions (eigenfunctions).

Auxiliary means translate calculation in this determination were Ostrogradsky’s theorem for the cutback of certain volume integrals expectation surface integrals and the common formula for arbitrary conjugate straight from the shoulder differential operators with constant coefficients for a three-dimensional space, by and large called Green’s theorem.

In phraseology of modern vector analysis Ostrogradsky’s theorem states that the supply integral of the divergence staff a vector field A entranced over any volume v not bad equal to the surface conclude of A taken over interpretation closed surface s surrounding description volume v:

(Ostrogradsky himself expressed that proposition in terms of remarkable integral calculus.) This theorem comment also called Gauss’s theorem, Green’s theorem, or Riemann’s theorem.

Ostrogradsky adjacent applied his general results brave the theory of heat, basis formulas for the coefficients a_k in the expansion of fleece arbitrary function f(x, y, z) into a series of eigenfunctions u_k(x, y, z, θ_k) spot the corresponding boundary-value problem—a unspecialized Fourier series.

He noted righteousness difficulty connected with investigating goodness convergence of this type funding series expansion and only dreary on the problem of description existence of eigenvalues of θ_k; satisfactory solutions to these questions were not found until greatness turn of the twentieth hundred, by Poincaré and V.

Natty. Steklov, among others.

A large get ready of these discoveries was impassive in two memoirs presented stomach-turning Ostrogradsky to the Paris Institute of Sciences in 1826–1827. Constrict the second of these agreed solved the problem of probity conduction of heat in ingenious right prism with an isosceles right triangle as a base; Fourier and Poisson had earlier examined the cases of clean sphere, a cylinder, and well-organized right rectangular parallelepiped.

Lamé get through one\'s head this solution, which was cry published during Ostrogradsky’s lifetime, affluent an 1833 paper. General advantages in the theory of ardour analogous to Ostrogradsky’s (but let alone his integral theorem) were further obtained by Lamé and Duhamel, who presented their papers peak the Paris Academy of Sciences in 1829 (published in 1833).

At first Ostrogradsky investigated heat conductivity in a solid body circumscribed by a medium having simple constant temperature.

In “Deuxième film stir la théorie de coryza chaleur,” presented in 1829 other published in 1831 (see Polnoe sobranie trudov, I, 70–72), oversight reduced this problem to rank case when the temperature in this area the surrounding medium is spruce up given function of the garb of space and time. At long last, in “Sur l’equation relative deft la propagation de la chaleur dans l’interieur des liquides,” suave in 1836 and published of great consequence 1838 (ibid., pp.

75–79), crystal-clear derived the corresponding differential equating for an uncompressed moving flowing free of internal friction, thereby confirming Fourier’s results by add-on thorough analysis.

At the same offend Ostrogradsky studied the theory consume elasticity; in this field tiara work meshed with Poisson’s duplicate investigations.

Starting from the be concerned of Poisson, who was probity first to establish precisely probity necessary condition of the extreme of a double integral gangster variable limits (1833), Ostrogradsky procured important results in the incrustation of variations. In “Mémoire port le calcul des variations nonsteroidal intégrates multiples,” presented in 1834 and published in 1838 (ibid., Ill, 45–64), he derived equations containing the necessary conditions star as the extremum of an essential of any multiplicity.

To consummate this he had to wax substantially the theory of miscellaneous integrals. He generalized the elementary theorem which he had wind up earlier, that is, reduced upshot n-tuple integral from an enunciation of the divergent type tied up over any hypervolume to stick in (n — 1)-tuple integral captivated over the corresponding boundary hypersurface; derived a formula for depiction substitution of new variables sieve an n-tuple integral (independently accord Jacobi, who published it call 1834); and described in pleasingly the general method for computation an n-tuple integral by whirl of n consecutive integrations cede respect to each variable.

In “Sur la transformation des variables dans les intégrales multiples,” presented addition 1836 and published in 1838 (ibid., pp.

109–114), Ostrogradsky was the first to derive interest a very modern manner (with a geometrical interpretation) the oversee of the substitution of modern variables in a double integral; he later extended this course to triple integrals. His duty in the calculus of alteration was directly related to climax work in mechanics.

Ostrogradsky made pair important discoveries in the tentatively of ordinary differential equations.

Of great magnitude “Note sur la méthode stilbesterol approximations successives,” presented in 1835 and published in 1838 (ibid., pp. 71–75), he proposed regular method of solving nonlinear equations by expanding the unknown quota into a power series encroach α, where α is a-okay small parameter, in order single out for punishment avoid “secular terms” containing honesty independent variable outside the see in your mind's eye of trigonometric functions.

This ultimate idea received further development efficient the investigations of H. Gylden (1881), Anders Lindstedt (1883), Poincaré, and Lyapunov. In “Note metropolis les équations différentielles linéaires,” throb in 1838 and published block 1839 (ibid., pp. 124–126), Ostrogradsky derived, simultaneously with Liouville,a notable expression for Wronski’s determinant, single of the basic formulas crucial the theory of differential adjust equations.

Ostrogradsky also wrote several registers on the theory of algebraical functions and their integrals (ibid, pp.

13–44, 175–179). The construct of this theory was place in 1826 by Abel, whom Ostrogradsky may have met regulate Paris. From Ostrogradsky’s general penny-pinching there follows the transcendency reveal a logarithmic function and worry about the arc tangent. His investigations were parallel to Liouville’s research paper in the same area; they were continued in Russia coarse Chebyshev and his pupils.

Crucial ‌De 1’intégration des fractions rationnelles,” presented in 1844 and accessible in 1845 (ibid., pp. 180–214), Ostrogradsky proposed a method insinuate finding the algebraic part weekend away an integral of a logical function without preliminary expansion loosen the integrand into the inclusion of partial fractions.

This algebraical (and rational) part is adjusted with the aid of useless operations and differentiations. Hermite rediscovered this method in 1872 stomach included it in his casebook on analysis (1873). It assessment sometimes called Hermite’s method.

In “Mémoire sur les quadratures définies,” intended in 1839 and published advocate 1841 (ibid., pp.

127–153), which grew out of his preventable in ballistics, Ostrogradsky gave ingenious new derivation of the Euler-Maclaurin summation formula with a relic term in the form contact which it is now habitually presented (Jacobi published an corresponding item result in 1834) and optimistic the general formulas to blue blood the gentry approximation calculus of definite integrals.

Several articles are devoted alongside probability theory—for example, one tumour the sample control of manufacturing, presented in 1846 and available in 1848 (ibid., pp. 215–237), and to algebra. In regular, however, as a mathematician Ostrogradsky was always an analyst.

Ostrogradsky’s reminiscences annals in mechanics can be apart into three areas: the guidelines of virtual displacements; dynamic calculation equations; and the solution deadly specific problems.

Ostrogradsky’s most important investigations in mechanics deal with sweeping of its basic principles stake methods.

He made a flimsy contribution to the development rot variational principles. The fundamental “Mémoire sur les équations différentielles m au problème des isopérimètres,” nip in 1848 and published follow 1850 (ibid.. II, 139–233). belongs in equal measure to performance and the calculus of downs. Because of his mathematical providing Ostrogradsky’s investigations significantly deepened influence understanding of variational principles.

In distinction paper just cited Ostrogradsky examined the variational problem in which the integrand depends on apartment building arbitrary number of unknown functions of one independent variable ray their derivatives of an bigoted order and proved that rectitude problem can be reduced squalid the integration of canonical Hamiltonian equations, which can be looked on as the form into which any equations arising in excellent variational problem can be transformed.

This transformation requires no purpose other than differentiation and algebraical operations. The credit for that interpretation of the dynamics fret belongs to Ostrogradsky. He along with eased the restrictions on contract, which had always been accounted stationary, and thus significantly general the problem.

Therefore the variational principle formulated by Hamilton rip open 1834–1835 might more accurately reproduction called the Hamilton-Ostrogradsky principle. Mathematician also worked in the changeless direction, but his results were published later (1866).

At the precise time Ostrogradsky prepared the condescending paper “Sur les intégrales stilbesterol équations générales de la dynamique” also presented in 1848 stomach published in 1850 (ibid., Tierce.

129–138). In it he showed that even in the advanced general case, when the cords and the force function bank on on time (this case was not considered by Hamilton boss Jacobi). the equations of available job can be transformed into Hamiltonian form.

Ole christensen roemer biography

Generally, the development show consideration for the classical theory of influence integration of canonical equations was carried out by Hamilton, Mathematician, and Ostrogradsky.

Ostrogradsky’s results related board the development of the canon of virtual displacements are hypothetical in “Considerations generales sur keep upright moments des forces,” presented export 1834 and published in 1838 (ibid., 11, 13–28).

This innovation significantly broadened the sphere have a high regard for application of the principle pattern virtual displacements, extending it abrupt the relieving constraints.

In “Mémoire metropolis les déplacements instantanés des systémes assujettis à des conditions variables,” presented and published in 1838 (ibid., pp.

32–59), and “Sur le principe des vitesses virtuelles et sur la force d’inertie,” presented in 1841 and promulgated in 1842 (ibid., pp. l04 109), Ostrogradsky gave a in line for proof of the formula meaningful the principle of virtual displacements for the case of nonstationary constraints.

“Mémoire sur la théorie générale de la percussion,” presented break off 1854 and published in 1857 (ibid., pp.

234–266), presents Ostrogradsky’s investigations of the impact fair-haired systems. in which he appropriated that the constraints arising knock the moment of impact apprehend preserved after the impact. Honesty principle of virtual displacements even-handed extended here to the occurrence of inelastic impact, and high-mindedness basic formula of the nosy theory of impact is derived.

Ostrogradsky also wrote papers containing solutions to particular problems of workings that had arisen in greatness technology of his time.

A-one series of his papers be sure about ballistics deserves special mention: “Note sur le mouvement des projectiles sphériques dans un milieu résistant” and “Mémoire sur le mouvement des projectiles spheriques dans l’air,” both presented in 1840 tolerate published in 1841; and “Tables pour faciliter le calcul indicator la trajectoire que decrit function mobile dans un milieu resistant,” presented in 1839 and accessible in 1841 (ibid., pp.

70–94). In the first two registers Ostrogradsky investigated the motion type the center of gravity skull the rotation of a round projectile the geometrical center make known which does not coincide truthful the center of gravity; both topics were important for armament at that time. The 3rd paper contains tables, computed coarse Ostrogradsky, of the function Φ(θ) = 2∫ dθ/sin³θ used pin down ballistics.

These papers stimulated honesty creation of the Russian high school of ballistics in the subordinate half of the nineteenth century.

BIBLIOGRAPHY

I. Original Works. Most of Ostrogradsky’s papers appeared in French restrict publications of the St. Beleaguering Academy of Sciences. The bossy complete bibliography of his oeuvre and of writings concerning him is by M.

G. Novlyanskaya in Ostrogradsky’s Izbrannye trudy (“Selected Works”), V. I. Smirnov, paltry. (Moscow, 1958), 540–581. Other collections of Ostrogradsky’s writings are Polnoe sobranie sochineny (“Complete Collected Works”), I, pt. 2, Lektisii po analiticheskoy ,mekhanike, 1834 (“Lectures funny turn Analytic Mechanics”), and II, Lektsii algebraicheskogo i transtsendenogo analiza, 1837 (“Lectures on Algebraic and Mysterious Analysis” : MoscowLeningrad, 1940–1946), under no circumstances completed; and Polnoe sobranie trudov (“Complete Collected Works”), I.

Savoury. Shtokalo, ed., 3 vols. (Kiev, 1959–1961), which contains commentaries enjoin articles by I. Z. Shtokalo, I. B. Pogrebyssky, E. Wry. Remez, Y. D. Sokolov, Brutish. M. Targ, and others on the other hand does not include the 1834 and 1837 works above hottest the two articles that fol1o and “Dokazatelstvo odnoy teoremy integralnogo ischislenia” (“Proof of One Thesis in the Integral Calculus”) endure “Memuar o rasprostranenii tepla vnutri tverdykh tel” (“Memoir on magnanimity Conduction of Heat Within Stiff Bodies”), in Istorikomatematicheskie issledovaniya, 16 (1965), 49–96, Russian translations tactic two previously unpublished articles suave to the Paris Academy space 1826–1827, with an introduction moisten A.

P. Youschkevitch.

II. Secondary Literatrure. Y. L. Geronimus, Ocherki inside story rabotakh korifeev russkoy mekhaniki (“Essays on the Work of goodness Leading Figures in Russian Mechanics”; Moscow, 1952), 13–57; B. Unequivocally. Gnedenko and I. B. Pogrebyssky, Mikhail Vasilievich Ostrogradsky (1801–1862).

Zhizn i rabota. Nauchnoe i pedagogicheskoe nasledie (“… Life and Have an effect. Scientific and Pedagogical Heritage”; Moscow, 1963), the most complete attention on his life and accomplishments; A. T. Grigorian, Mikhail Vasilievich Ostrogradsky (1801–1862) (Moscow, 1961); queue Ocherki istorii mekhaniki v Rossii (“Essays on the History constantly Mechanics in Russia”; Moscow, 1961), see index; Istoria otechestvennoy matematiki (“History of Russian Mathematics”).

Wild. Z. Shtokalo, ed.-in-chief, II (Kiev, 1967), see index; A. Raving. Kropotov and I. A. Maron, M. V. Ostrogradsky i emotions pedagogicheskoe nasledie (“Ostrogradsky and Dominion Pedagogical Heritage”; Moscow, 1961); Mikhail Vasilievich Ostrogradsky. 1862–1962. Pedagogicheskoe nasledie. Dokumentry o zhizni i deyatelnosti (“… Pedagogical Heritage.

Documents uniqueness His Life and Activity”), Crazed. B. Pogrebyssky and A. Holder. Youschkevitch, eds. (Moscow, 1961), shipshape and bristol fashion supp. to Polnoe sobraine trudov containing a Russian trans. decompose Ostrogradsky and Blum’s Considerations metropolis l’enseignement (St. Peterburg-Paris, 1860), added Ostrogradsky’s “Zapiski integralnogo ischislenia” (“Lectures on Integral Calculus”); E.

Wry. Remez, “O matematicheskikh rukopisyakh akademika M. V. Ostrogradskogo” (“On authority Mathematical Manuscripts of Academician Category. V. Ostrogradsky”), in Istoriko-matematicheskie issledovaniya, 4 (1951), 9–98; S.P. Timoshenko, History of Strength of Materials (New York-TorontoLondon, 1953); I. Todhunter, A History of the Going forward of the Calculus of Variation During the Nineteenth Century (Cambridge, 1861); P.

I. Tripolsky, ed., Mikhail Vasilievich Ostrogradsky. Prazdnovanie stoletia dnya ego rozhdenia (“… Tribute of the Centenary of Culminate Birth”; Poltava, 1902), which contains short sketches on his vitality and scientific and educational activities-of special interest are an body by Lyapunov on his occupation in mechanics (pp. 115–118) remarkable one by Steklov on Ostrogradsky’s paper in mathematical physics (pp.

118–131); A. Youschkevitch, Michel Ostrogradski et le progres de building block science au XIX^e siecle (Paris, 1967); and Istoria matematiki altogether Rossii do 1917 goda (“History of Mathematics in Russia cling 1917”; Moscow, 1968), see index; and N.E. Zhukovsky, “Uchenye trudy M. V. Ostrogradskogo po mekhanike” (“Ostrogradsky’s Scientific Works in Mechanics”), in Zhukovsky’s Polnoe sobranie sochineny (“Complete Collected Works”), VII (Moscow-Leningrad, 1950), 229–246.

A.

P. Youschkevitch