# Yurii Alekseevich Mitropolskii

### Quick Info

Charnyshivka, Shyshats'kyi, Poltava gubernia, Ukraine

Kiev, Ukraine

**Yurii Alekseevich Mitropolskiy**was a Ukrainian mathematician known for his contributions to dynamical systems and nonlinear oscillations.

### Biography

**Yurii Alekseevich Mitropolskii**'s name is often transliterated as Mitropolsky or Mytropolsky and occasionally as Mytropolskiy or Mitropolskiy. His father, Aleksei Savvich Mitropolskii, had attended St Petersburg University but was called up for military duty in 1914. When his son Yurii Alekseevich was born, Aleksei Savvich was serving at the front. After he was demobbed in 1919 he went with his family to Kiev where Yurii Alekseevich was brought up from the age of two. As a child he worked in a factory in Kiev. He entered the Faculty of Mechanics and Mathematics of Shevchenko Kiev State University in 1938 but on 22 June 1941 the German armies invaded their former allies pushing rapidly east into Soviet lands. At first their main advance was aimed towards Moscow, but by August they made a strong push in the south deep into the Ukraine heading towards Kiev. The University was evacuated from Kiev before the German troops reached the city and Mitropolskii was sent to the front. He was recalled from the front to continue his education at the Department of Physics and Mathematics at Kazakh University in Alma-Ata (renamed Almaty in 1991). Alma-Ata is the Soviet version of the Kazakh name Almaty for the capital of Kazakhstan, meaning "Father of Apples". It took this name in 1921 having previously been named Verny. Kazakh Al-Farabi State University was very new when Mitropolskii studied there, the University being founded in 1934.

Mitropolskii graduated from the Kazakh University in 1942 after studying there for six months. After graduating he attended the Ryazan Military Artillery School and then, from 1943 until the end of the war, he was sent to the front where he commanded an artillery intelligence platoon. The authors of [41] write for Mitropolskii's ninetieth birthday:-

For services in battle, he was awarded two Orders of the Red Star and military medals. Mitropolskii retains vivid memories of wartime, and he still remembers his comrades in arms and even details of the army life quite well.He continued his military service until he was demobbed in 1946 when he began research at the Institute of Constructive Mechanics of the Academy of Sciences of the Ukraine, working under Nikolai Nikolaevich Bogolyubov. He was awarded his Candidate's Degree (equivalent to a Ph.D.) in 1948 for his dissertation on the problem of resonance phenomena in non-linear oscillatory systems with slowly varying parameters. His approach to the problem used the Krylov-Bogolyubov asymptotic methods. He continued to work for his doctorate (equivalent to the habilitation) and he was awarded this in 1951 for his thesis

*Slow processes in non-linear oscillatory systems with many degrees of freedom*. In this impressive piece of work he studied problems of non-linear mechanics and mathematical physics which involved investigating non-stationary phenomena in non-linear oscillatory systems. He moved to the Institute of Mathematics of the Academy of Sciences of the Ukraine in 1951 and, two years later, he was appointed head of the Department of Mathematical Physics and Non-linear Oscillation Theory.

From 1951 Mitropolskii taught in the Faculty of Mechanics and Mathematics at Kiev University, where he was named as professor in 1954, and continued teaching there when made Director of the Institute of Mathematics in 1958. He held the post of Director of the Institute for 30 years, expanding the work of the Institute. Volodymyr Petryshyn writes in [36]:-

During Yu Mitropolskii's directorship (1958-88), the Institute experienced a great expansion in research personnel and mathematical disciplines, and an improvement in the quality of research.Anatoly Samoilenko was a student of Mitropolskii's who obtained his Ph.D. in 1963. Following this, he worked with Mitropolskii on many joint mathematical projects and, when Mitropolskii retired from the directorship of the Institute in 1988, Samoilenko took over the directorship.

In [37] Volodymyr Petryshyn summarises Mitropolskii's work as follows:-

Mitropolskii has made major contributions to the theory of oscillations and nonlinear mechanics as well as the qualitative theory of differential equations. He further developed asymptotic methods and applied them to the solution of practical problems. He extended the Krylov-Bogolyubov symbolic method to nonlinear systems and extended asymptotic methods in the theory of nonlinear mechanics. Using a method of successive substitutes, he constructed a general solution for a system of nonlinear equations and studied its behaviour in the neighbourhood of the quasi-periodic solution. He also successfully applied the averaging method to the study of oscillating systems with slowly varying parameters.The authors of [9] list seven main areas in which Mitropolskii made significant contributions:-

*the creation and mathematical justification of algorithms for constructing asymptotic expansions for non-linear differential equations describing non-stationary oscillatory processes*;*the development of a method for investigating monofrequency processes in oscillatory systems*;*the investigation of systems of non-linear differential equations describing oscillatory processes in gyroscopic systems and strongly non-linear systems*;*the development of the theory of integral manifolds in non-linear mechanics and the consideration of related questions that arise on stability of motion*;*the development of the averaging method for equations with slowly varying parameters, as well as for equations with non-differentiable and discontinuous right-hand sides, for equations with delayed argument, for equations with random perturbations, and for partial differential equations and equations in functional spaces*;*the development of the method of accelerated convergence in problems of non-linear mechanics*;*the development of the theory of reducibility in linear differential equations with quasi-periodic coefficients, and other equations*.

The present book is the fourth or fifth major treatise published in recent years by Soviet scientists on the general topic of non-linear oscillations, which serves to indicate the great value which is attached in the USSR to this general topic. The general program of the book is not too far from the program of the 1937 Krylov-Bogolyubov monograph [Introduction to non-linear mechanics (1937)]. However, although the book is addressed primarily to physicists and engineers, its mathematical treatment is most careful, which was by no means the case with the 1937 monograph. The book is also much more orderly and most readable: an excellent contribution in every respect.This work was to lead to further advances by the Kiev school, in particular they applied asymptotic methods to partial and functional differential equations. An English translation of the second Russian edition of the book (containing an additional chapter on single-frequency oscillations in systems with many degrees of freedom) appeared as

*Asymptotic methods in the theory of non-linear oscillations*in 1961. A French translation appeared in 1962 with a German translation three years later. The method developed by the authors and presented in this and later editions of the monograph have come to be known as the KBM method (Krylov-Bogolyubov-Mitropolskii). This book was the first of many books written by Mitropolskii, the majority co-authored with his former doctoral students. The authors of [8] list 31 monographs published by Mitropolskii between 1955 and 2005. Among the single authored texts we mention:

*Nonstationary processes in non-linear oscillatory systems*(1955);

*Problems in the asymptotic theory of non-stationary oscillations*(1964);

*Lectures on the method of averaging in non-linear mechanics*(1966);

*The method of averaging in nonlinear mechanics*(1971);

*Nonlinear mechanics. Asymptotic methods*(1995);

*Non-linear mechanics. Monofrequency oscillation*(1997); and

*Methods of non-linear mechanics. A first textbook*(2005). A reviewer of the 1964 monograph writes:-

The book is written for readers interested in the application of the techniques described. Asymptotic solutions of differential equations are worked out in great detail, the author always being willing to go the second mile with the reader in obtaining the inherently complicated formulas that arise. A large number of physical problems are presented, again in careful and lengthy detail.Let us also quote from the Preface of the 1971 monograph:-

We deal with the method of averaging in nonlinear mechanics. We include numerous results of further development and generalization of the basic ideas of N N Bogolyubov. We give various algorithms, schemes and rules for constructing approximate solutions of equations with small and large parameters, and obtain examples which in many cases graphically illustrate the effectiveness of the method of averaging and the breadth of its application to various problems which are, at first glance, very disparate. The theorems that we include reveal the depth and mathematical rigour of the method of averaging. We discuss the basic trends and developments of the method of averaging, and as illustrations we give typical examples of nonlinear oscillatory systems, revealing the effectiveness of the method.Among the many co-authored works we mention

*Lectures on the application of asymptotic methods to the solution of partial differential equations*(1968) co-authored with his former student Boris Illich Moseenkov,

*Lectures on the methods of integral manifolds*(1968) co-authored with his former student Olga Borisovna Lykova,

*Lectures on the theory of oscillation of systems with lag*(1969) co-authored with his former student Dmitrii Ivanovich Martynyuk,

*Asymptotic solutions of partial differential equations*(1976) co-authored with his former student Boris Illich Moseenkov,

*Periodic and quasiperiodic oscillations of systems with lag*(1979) also co-authored with D I Martynyuk,

*Mathematical justification of asymptotic methods of nonlinear mechanics*(1983) co-authored with his former student Grigorii Petrovich Khoma,

*Group-theoretic approach in asymptotic methods of nonlinear mechanics*(1988) co-authored with his former student Aleksey Konstantinovich Lopatin, and

*Asymptotic methods for investigating quasiwave equations of hyperbolic type*(1991) co-authored with his former students G P Khoma and Miron Ivanovich Gromyak. We give three examples of complimentary comments from reviewers of these texts:-

*The book constitutes a welcome addition to the literature on this subject*.*This is an excellent monograph whose main purpose is to present a mathematical justification of the method of averaging and in particular of the Krylov-Bogolyubov asymptotic method*.*The book is well written and it may be recommended to researchers and students interested in oscillatory processes*.

We should also mention his important contribution to mathematics as an editor of several different journals. Some of these were Ukrainian journals such as the differential equations journal

*Differentsial'nye Uravneniya*, while others were international journals such as the

*International Journal of Nonlinear Sciences and Numerical Simulations*, the journal

*Nonlinear Analysis*, the journal

*Nonlinear Dynamics*, and the

*International Journal of Nonlinear Mechanics*. He was also interested in the history of mathematics and served on the editorial board of the

*History of the Ukrainian Academy of Sciences*and

*Essays on the development of mathematics in the USSR*. To illustrate this interest let us mention his book

*The Institute of Mathematics: Academy of Sciences of the Ukrainian SSR*written jointly with V V Strok and published in 1988.

Mitropolskii was elected to the Academy of Sciences of the Ukraine in 1961, to the Bologna Academy of Sciences (1971), and to the Academy of Sciences of the USSR in 1984. He was also honoured by the award of the A M Lyapunov Gold Medal in 1987. He was a speaker at the International Congress for Mathematicians in Edinburgh in 1958, in Stockholm in 1962, in Moscow in 1966, in Nice in 1970, in Vancouver in 1974, in Warsaw in 1983, in Berkeley in 1986, and in Kyoto in 1990. In 1965 he was awarded the Lenin Prize:-

... for his outstanding achievements in the theory of nonlinear differential equations and nonlinear oscillations.He was also honoured with the award of the Krylov Prize (1969), the Bogolyubov Prize (1994), State Prizes of the Ukraine (1980 and 1996), and the Lyapunov Gold Medal (1986):-

... for the development of asymptotic methods in nonlinear mechanics.For the same work he was awarded the Silver Medal of the Czech Academy of Sciences (1978):-

... for services to science and mankind.He was made a Hero of Ukraine in January 2007 and, on the occasion of his ninetieth birthday, he was presented with the V I Vernadskii Gold Medal by the President of the National Ukrainian Academy of Sciences.

As to his personal characteristics, his colleagues write of his:-

... extraordinary creative energy, vigour, and optimism.

### References (show)

- A N Bogoljubov (ed.),
*Yurii Alekseevich Mitropolskii: a bibliography : Compiled by M N Kreknina*(Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 1977). - V K Kraineva and Ya A Matviishin,
*Yurii Alekseevich Mitropolskii : With an introduction by A N Bogolyubov, O B Lykova and A M Samoilenko*, Biobibliography of Scientists of the Ukrainian SSR 'Naukova Dumka' (Kiev, 1987). - 60th birthday of the Ukrainian SSR Academy of Science's Academician Yu A Mitropolskii (Russian),
*Godishnik Vissh. Uchebn. Zaved. Tekhn. Mekh.***11**(1) (1976), 7-9. - Academician Jurii Olekseeovich Mitropolskii on the occasion of his sixtieth birthday (Bulgarian),
*Teoret. i Priloz. Meh.***8**(1) (1977), 9-10. - Academician Yu A Mitropolskii (on the occasion of his seventieth birthday) (Russian),
*Ukrain. Mat. Zh.***39**(1) (1987), 3-4. - Academician Yu A Mitropolskii - hero of socialist labour (Russian),
*Vestnik Akad. Nauk SSSR*(5) (1987), 131-132. - Academician Yu O Mitropolskii (on the occasion of his eightieth birthday) (Ukrainian),
*Ukrain. Mat. Zh.***49**(1) (1997), 3-4. - V I Arnold, V S Vladimirov, VV Kozlov, E F Mishchenko, Yu S Osipov, B E Paton, A N Sisakyan, A D Sukhanov, L D Faddeev, and K V Frolov, Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday) (Russian),
*Uspekhi Mat. Nauk***62**(4)(376) (2007), 179-185. - V I Arnold, V S Vladimirov, VV Kozlov, E F Mishchenko, Yu S Osipov, B E Paton, A N Sisakyan, A D Sukhanov, L D Faddeev, and K V Frolov, Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday),
*Russian Math. Surveys***62**(4) (2007), 829-835. - D Bainov, Jurii Alekseevich Mitropolskii, on the occasion of his 60th birthday (Bulgarian),
*Fiz.-Mat. Spis. B'lgar. Akad. Nauk.***20**(53) (2) (1977), 174-175. - N N Bogoljubov and V S Koroljuk, Ju O Mitropolskii's studies in the field of nonlinear oscillation theory (Russian),
*Ukrain. Mat. Z.***29**(1) (1977), 3-14, 141. - N N Bogoljubov, V S Koroljuk and A M Samoilenko, Jurii Alekseevich Mitropolskii (on the occasion of his sixtieth birthday) (Russian),
*Uspekhi Mat. Nauk***32**(1)(193) (1977), 217-228. - N N Bogolyubov, E F Mishchenko and A M Samoilenko, Yurii Alekseevich Mitropolskii (on the occasion of his seventieth birthday) (Russian),
*Uspekhi Mat. Nauk***42**4(256) (1987), 193-195. - M O Bogolyubov and O B Likova, Yurii Oleksiiovich Mitropolskii (on the occasion of his eightieth birthday) (Ukrainian), in
*The Institute of Mathematics. Outlines of its development*(Ukrainian), (Natsional. Akad. Nauk Ukraini, Inst. Mat., Kiev, 1997), 147-155. - S Djadkov, Yu A Mitropolsky is sixty,
*Acta Tech. CSAV***21**(6) (1976), 621-622. - K V Frolov, E F Mishchenko, O A Oleinik, Yu S Osipov, A, M Samoilenko and V S Vladimirov, Yurii Alekseevich Mitropol'skii (on his eightieth birthday) (Russian),
*Uspekhi Mat. Nauk***52**(1) (1997), 237-239. - K V Frolov, E F Mishchenko, O A Oleinik, Yu S Osipov, A, M Samoilenko and V S Vladimirov, Yurii Alekseevich Mitropol'skii (on his eightieth birthday),
*Russian Math. Surveys***52**(1) (1997), 237-239. - N P Erugin, V S Koroljuk and O B Lykova, Jurii Alekseevich Mitropolskii (on the occasion of his 60th birthday) (Russian),
*Differencial'nye Uravnenija***13**(1) (1977), 177-184. - N P Erugin, V S Koroljuk and O B Lykova, Yurii Alekseevich Mitropolskii (on the occasion of his seventieth birthday) (Russian),
*Differentsial'nye Uravneniya***23**(1) (1987), 3-9. - N P Erugin, Ju D Sokolov, S F Fescenko and O B Lykova, Jurii Alekseevich Mitropolskii (Russian),
*Differencial'nye Uravnenija***3**(1967), 158-166. - I V Gaishun, V A Il'in, N A Izobov, V S Korolyuk, V N Koshlyakov, V F Kravchenko, I A Lukovskii, A A Martynyuk, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasilev, Yurii Alekseevich Mitropolskii (Russian),
*Differentsial'nye Uravneniya***44**(12) (2008), 1711-1713. - I V Gaishun, V A Il'in, N A Izobov, V S Korolyuk, V N Koshlyakov, V F Kravchenko, I A Lukovskii, A A Martynyuk, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasilev, Yurii Alekseevich Mitropolskii,
*Differential Equations***44**(12) (2008), 1776-1778. - I V Gaishun, N A Izobov, V A Il'in, V S Korolyuk, V N Koshlyakov, I A Lukovskii, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasileva, Yurii Alekseevich Mitropolskii (A Tribute in Honor of His Ninetieth Birthday) (Russian),
*Differentsial'nye Uravneniya***43**(1) (2007), 1-9. - I V Gaishun, N A Izobov, V A Il'in, V S Korolyuk, V N Koshlyakov, I A Lukovskii, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasileva, Yurii Alekseevich Mitropolskii (A Tribute in Honor of His Ninetieth Birthday),
*Differential Equations***43**(1) (2007), 3-10. - V M Gluskov, O S Parasjuk, V S Koroljuk and O B Lykova, Jurii Alekseevich Mitropolskii (Russian),
*Ukrain. Mat. Z.***19**(1) (1967), 3-8. - L Hatvani, Yurii Alekseevich Mitropolskii, founder of ICNO, in
*Proceedings of the Eleventh International Conference on Nonlinear Oscillations*(Budapest, 1987), 14-17. - V A Il'in, N A Izobov, A A Martynyuk and A M Samoilenko, Yurii Alekseevich Mitropolskii (on the occasion of his eightieth birthday) (Russian),
*Differ. Uravn.***33**(1) (1997), 3-5. - V A Il'in, N A Izobov, A A Martynyuk and A M Samoilenko, Yurii Alekseevich Mitropolskii (on the occasion of his eightieth birthday),
*Differential Equations***33**(1) (1997), 1-3. - Juri Alekseevich Mitropolskii (on the occasion of his sixtieth birthday) (Russian),
*Mat. Fiz. Vyp.***22**(1977), 3-4. - J Kurzweil, On the sixtieth birthday of Yu A Mitropolskii (Czech),
*Pokroky Mat. Fyz. Astronom.***22**(1) (1977), 40-41. - V Lakshmikantham, A A Martynyuk and J H Dshalalow, Personage in science : Academician Yu A Mitropolskii,
*Nonlinear Dyn. Syst. Theory***6**(4) (2006), 309-318. - O Limarchenko and J-H He, Personage in science : Academician Yury Mitropolsky,
*Int. J. Nonlinear Sci. Numer. Simul.***1**(1) (2000), 3-5. - Obituary for Yurii Alexeevich Mitropolskii (1917-2008),
*Nonlinear Dyn. Syst. Theory***8**(4) (2008), 407-410. - On the 80th Birthday of Academician Yu A Mitropolskii (Russian),
*Ukrainskii Matematychnii Zhurnal***49**(1) (1997), 3-4. - On the 80th Birthday of Academician Yu A Mitropolskii,
*Ukrainian Mathematical Journal***49**(1) (1997), 1-2. - V Petryshyn, Mathematics,
*Encyclopaedia of Ukraine*(Toronto-Buffalo-London, 1993), 339. - V Petryshyn, Mytropolsky, Yurii,
*Encyclopaedia of Ukraine*(Toronto-Buffalo-London, 1993). - A M Samoilenko et al., Yurii Oleksiiovich Mitropolskii (on the occasion of his ninetieth birthday) (Ukrainian),
*Neliniini Koliv.***10**(1) (2007), 4-5. - A M Samoilenko et al., Yurii Oleksiiovich Mitropolskii (on the occasion of his ninetieth birthday),
*Nonlinear Oscil. (N. Y.)***10**(1) (2007), 1-3. - A M Samoilenko, Yu M Berezanskyi, V S Korolyuk, V M Koshlyakov, I O Lukovskyi, O M Sharkovskyi, M L Horbachuk, V L Makarov, M O Perestyuk, Yu I Samoilenko, O O Stepanets, P M Tamrazov, Yu Yu Trokhymchuk and V V Sharko, On the ninetieth birthday of Yurii Alekseevich Mitropolskii (Russian),
*Ukrainskii Matematychnii Zhurnal***59**(2) (2007), 147-151. - A M Samoilenko, Yu M Berezanskyi, V S Korolyuk, V M Koshlyakov, I O Lukovskyi, O M Sharkovskyi, M L Horbachuk, V L Makarov, M O Perestyuk, Yu I Samoilenko, O O Stepanets, P M Tamrazov, Yu Yu Trokhymchuk and V V Sharko, On the ninetieth birthday of Yurii Alekseevich Mitropolskii,
*Ukrainian Mathematical Journal***59**(2) (2007), 153-157. - A M Samoilenko and V G Kolomiets, On Yu O Mitropolskii's contribution to the development of the asymptotic methods of nonlinear mechanics (Ukrainian),
*Ukrain. Mat. Zh.***49**(1) (1997), 5-10. - A M Samoilenko and O B Lykova, Development of the methods of nonlinear mechanics in the works of Yu A Mitropolskii (Russian),
*Ukrain. Mat. Zh.***39**(1) (1987), 5-13. - N Stojanov and D Bainov, Academician Jurii Alekseevich Mitropolskii (on the occasion of his 60th birthday) (Bulgarian),
*Godishnik Vissh. Uchebn. Zaved. Prilozhna Mat.***12**(1) (1976), 9-11. - The awarding of the A M Lyapunov Gold Medal to Yu A Mitropolskii (Russian),
*Vestnik Akad. Nauk SSSR*(2) (1987), 136. - The studies of Yu A Mitropolskii on the field of the theory of nonlinear oscillations and the theory of nonlinear differential equations (Russian), in
*Problems of the asymptotic theory of nonlinear oscillations***275**'Naukova Dumka' (Kiev, 1977), 7-14. - A B Vasilev, I V Gaishun, N A Izobov et al., Yurii Alekseevich Mitropolskii (Russian),
*Differ. Uravn.***44**(12) (2008), 1711-1713. - A B Vasilev, I V Gaishun, N A Izobov et al., Yurii Alekseevich Mitropolskii,
*Differ. Equ.***44**(12) (2008), 1776-1778. - A B Vasileva et al., Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday) (Russian),
*Differ. Uravn.***43**(1) (2007), 3-10. - A B Vasileva et al., Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday),
*Differ. Equ.***43**(1) (2007), 1-9. - V S Vladimirov, E F Mishchenko, O A Oleinik, Yu S Osipov, A M Samoilenko and K V Frolov, Yurii Alekseevich Mitropolskii (on the occasion of his eightieth birthday) (Russian),
*Uspekhi Mat. Nauk***52**1(313) (1997), 237-239. - Yurii Alekseevich Mitropolskii (on the occasion of his 75th birthday) (Russian),
*Ukrain. Mat. Zh.***44**(1) (1992), 3-4. - Yurii Alekseevich Mitropolskii (on the occasion of his seventieth birthday) (Russian),
*Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat.*(1) (1987), 88-89. - Yurii Aleksiiovich Mitropolskii (on the occasion of his sixtieth birthday) (Ukrainian), V
*isnik Kiev. Univ. Ser. Mat. Mekh. No.***19**(1977), 143-144. - Yurii Oleksiiovich Mitropolskii (Ukrainian),
*Neliniini Koliv.***11**(3) (2008), 292. - Yurii Oleksiiovich Mitropolskii,
*Nonlinear Oscil. (N. Y.)***11**(3) (2008), 305-306. - Yurii Oleksiiovich Mitropolskii (Ukrainian),
*Ukrain. Mat. Zh.***60**(9) (2008), 1155-1156. - Yurii Oleksiiovich Mitropolskii,
*Ukrainian Math. J.***60**(8) (2008), 1347-1348.

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Written by J J O'Connor and E F Robertson

Last Update November 2010

Last Update November 2010