Viktor Grigorjevich Veretennikov

1938-2008

P.S.Krasilnikov

Moscow Aviation Institute

Volokolamskoe shosse, 4, Moscow, A-80, GSP-3, 125993, Russia

Victor Grigorjevich Veretennikov - Professor, Corresponding Member of the Russian Academy of Science, the honored worker of a science of the Russian Federation, the known scientist in the field of stability theory and theory of nonlinear oscillations. His scientific results are of great importance to the theory of Lyapunov`s critical cases, theory of periodic and quasi-periodic motions, analytical mechanics.

Victor Grigorjevich was born on February, 25th, 1938 in Khabarovsk. He led children's and teenager years in Chistopol of Tatarstan where he leaved high school with a silver medal. He entered Kazan Aviation Institute which was finished in 1961. At this time, the Kazan scientific school of analytical mechanics was one of the leading scientific schools of Russia. An outstanding scientist, N.G.Chetaev worked a long time here. He united around of itself the talented youth researchers which are interesting in analytical mechanics, nonlinear theory of oscillations and stability theory. Keen by scientific researches, Victor Grigorjevich get a postgraduate education of MAI. His supervisor of studies was the professor G.V.Kamenkov which was the eminent Russian scientist. G.V.Kamenkov headed нбI as a rector at this time. The problems of stability in critical cases became a subject of his researches for many years. In the end of the 60-th years, he stayed at the theoretical mechanics department after defending the Ph. D dissertation. He filled a position of a senior teacher. He received a Doctor of Science degree when he was 34 years old and he became a head of "Theoretical mechanics" department. Since 1977, Victor Grigorjevich was a dean of faculty . It is necessary to tell, that becoming of faculty was not simply. Not all people in нбI understood the importance of applied mathematics for a technical university; moreover, some of eminent mathematicians of Russia criticized the division of mathematics into two domains ("pure" and "applied") sharply. In Academy of sciences of the USSR, this laded to hot discussion on the relationship between pure and applied mathematics. It now, after many years, it became understandable that just applied researches in mathematics generated a revolutionary scientific field such as the information technologies which are a foundation of researches in many areas of human activity. Victor Grigorjevich has worked hard for becoming of faculty and he was supported by rector of MAI.

Victor Grigorjevich has carried out the reforms in the field of scientific work and teaching practices at the faculty. He has been assured that educational process in higher school is inconceivable without active scientific work of teachers. He invited the eminent scientists such as academicians V.S.Pugachev, R.F.Ganiev, V.M.Matrosov, prof. U.G.Pirumov and prof. A.P.Markeev to work on faculty. Involving the talented young men into postgraduate education, demanding the defending of dissertations in time, he increased a level of scientific researches sharply. He was considerate towards the teachers which were engaged in scientific work actively, was interested in their results and supported them towards career growth. Under his management, 13 pupils received Ph.D degree and 4 pupils received Doctor of Science degree. Two scientific schools were created at the faculty. Professor V.G.Veretennikov was a Head of scientific school of analytical mechanics and nonlinear oscillations, professor U.G.Pirumov headed school of mechanics of gas and liquid. Both schools had many scientific results and pupils. As a result, these schools have led to creation two dissertation Committee for defence of thesis of Doctor of Science and Ph.D.

Victor Grigorjevich knew how to work with people; he was very gifted for understanding of people. After first meeting, he had understood the temper of person and his talent immediately. He was mistaken seldom. He based on honest and talent persons, sharply counteracted to intrigues which took place at faculty in the beginning of his career.

Since 1992, Victor Grigorjevich had been a vice-rector of MAI. In 2001 year, he became the government prize-winner in since and technology areas. In 2003 year, he became the member-correspondents of RAS, received an award of a labor Red banner and the Belgian award "Commandeur".

His first work was devoted to forced oscillations of system with two degrees of freedom [1]. After this, he started to investigate the theory of stability of stationary motions. He considered the stability problem in critical case of three pairs of pure imaginary eigenvalues. When resonance does not exist, he got the criterion of asymptotic stability as a third approximation. As distinguished from A.M.Molchanov considering the common case of q (q³3) pairs of pure imaginary eigenvalues, V.G.Veretennikov investigates the stability when the noncritical variables were taken into the consideration and constructs Lyapunov's function V = r exp[-Nu(z)] in explicit form. Victor Grigorjevich considered also some case when the conditions of his theorem and Molchanov's theorem are not realized. These results were applied to investigate the stability of gyro horizon by help of precession and nutation equations [4].

The contributions [3, 18] are devoted to further researches of critical cases. V.G.Veretennikov considered the common problem of transformation of nonlinear quasiperiodic systems containing critical and noncritical variables. Following to A.M.Lyapunov and G.V.Kamenkov, he separate the problems of Lyapunov's critical cases into two subcases. The first subcase called as an inessential special case contains the differential equations which stability investigated via finite approximation. The second subcase called as essential special case needs the researches of all forms of right hand sides for equations. In first case, Victor Grigorjevich proved that equations of perturbed motions can be transform to such form when critical and noncritical variables are separated partly. New equations in critical variables are independent of time and noncritical variables up to members of order N inclusive. In this case, the stability problem is equivalent to stability problem of autonomous N-approximation system containing the critical variables only. In the most common critical case,љ he showed, by means of this transformation, that stability problem of quasi-periodic system is reduced to stability problem of autonomous system in the critical case of (p+2q) zero eigenvalues where p and q are the numbers of zero roots and pure imaginary roots of characteristic equation for quasi-periodic system.

V.G.Veretennikov considered the essential special case [9,10] also. It takes place when the set of nonlinear expressions for critical equations are vanish on the manifold of solutions for some system of partial differential equations. He proved the theorem of the existence of the holomorphic solutions for such PDE by means of the generalization some Kamenkov's theorem. With the help of this theorem, he construct the convergent mapping which change the system of essential case to special form and prove the stability of nonperturbed motion.

The next works [7, 8, 18] are devoted to the stability in the cases closed to critical ones. This nearness is defined by smallness of real parts of eigenvalues that implies a little areas of stability (instability). Moreover, these areas can be as much as small. Here the stability is defined by G.V.Kamenkov, i.e. there is exists some closed area G containing the origin such that if initial conditions belong to G, the unperturbed motions belong G also. V.G.Veretennikov constructs the mapping which reduce the stability problem to investigation of N-approximation system. If G satisfies to some condition of asymptotic behavior inside of G, he proved the theorem of the existence of G by means ofљ Lyapunov's function. With the help of this theorem, he investigated the case closed to critical one which contains n pairs of complex conjugate eigenvalues. He researched the stability problem in essential special case also.

The contributions [15, 19, 20, 21] are devoted to further investigation of stability by G.V.Kamenkov. With the help of v-functions, he developed the methods and algorithms of estimates for domains of stability, attractors, integral vortexes and time cross-sections of vortexes, invariant sets also. He got necessary and sufficient conditions of stability on the sets. At this time he headed a lot of scientific works in MAI which were devoted to problems of flow mechanics, aerodynamics of parachute systems, dynamics of gyroscopic devices. He used the developed methods to investigate these problems.

Lately he began to study the fundamental concepts of analytical mechanics and the correlations of ones with the geometry, theory of differential equations, functional analysis, physics, technics. The results of the investigations are published in two monographers which were written with V.A.Sinitsyn together [32, 36], and in the articles [26-28, 30, 31, 33-38]. The area of the researches is rather extensive. We shall specify only two directions of researches. The first, using N.G.Chetaev's interpretation of mechanical principle of release from constraints, he has proved the extension of d'Alembert-Lagrange principle to any systems of ordinary differential equations with constraints (DAE-equations). The second, investigating the integral principles of mechanics, he pays main attention to variation methods with synchronous, asynchronous and Helmholtz's variations, to creation the new integral equalities of analytical mechanics.

Didn't fulfill oneself, Victor Grigorjevich pass away. Light memory of him remains in the hearts of many people who knew him.

 

Main publications list of Prof. V.G.Veretennikov

1.         V.G.Veretennikov. Investigation of forced oscillations for nonlinear systems with two degree of freedom. Proceedings of UDN, Series theoretical mechanics, v. 15, issue 3, 1966 (in Russian).

2.         V.G.Veretennikov. Investigation of stability in the case of three pairs of pure eigenvalues. Proceedings of UDN, Series theoretical mechanics, v. 15, issue 3, 1966 (in Russian).

3.         V. G. Veretennikov. љOn the stability of almost-periodic motions. PMM, v. 32, issue 1, 1968 (in Russian).

4.         V. G. Veretennikov. On the stabilization of neutral systems. Proceedings of UDN, Series theoretical mechanics, v. 27, issue 5, 1968 (in Russian).

5.         V. G. Veretennikov.љ Construction of solutions of quasilinear non-autonomous systems at the resonances. PMM, v. 33, issue 6, 1969 (in Russian).

6.         V. G. Veretennikov.љ Domains of stability in the cases closed to critical ones. PMM, v. 35, issue 1, 1971 (in Russian).

7.         V. G. Veretennikov. On the investigation of stability for systems closed to critical case. PMM, v. 35, issue 6, 1971 (in Russian).

8.         V. G. Veretennikov. Investigation of stability for systems closed to critical case. In the collection of contributions "Problems of mechanics of controlled motions", Perm, State University of Perm, issue 1, 1972 (in Russian).

9.         V. G. Veretennikov. Investigation of stability of quasi-periodic motions in transcendental critical cases. In "Problems of analytical mechanics, stability theory and control", "Nauka", Moscow, 1975 (in Russian).

10.      V. G. Veretennikov. About one transformation in stability theory. In the collection of contributions "Some questions of mechanics", issue 321, MAI, Moscow, 1975 (in Russian).

11.      V. G. Veretennikov, V.N. Seregin. On the investigation of oscillations for quaisilinear systems with quasi-periodic coefficients. PMM, v. 43, issue 6, 1979 (in Russian).

12.      V. G. Veretennikov, A.P. Markeev. Investigation of stability of nonlinear systems. MAI, Moscow, 1979 (in Russian).

13.      V. G. Veretennikov, A.I. Gurin. Some questions of dynamic for systems with cyclic coordinates. MAI, Moscow, 1980 (in Russian).

14.      V. G. Veretennikov, S.V. Medvedev.љ Normal forms of perturbed equations with quasi-periodic coefficients. In the collection of contributions "Analytical methods of mechanics in dynamic aircraft problems" MAI, Moscow, 1982 (in Russian).

15.      V. G. Veretennikov, V.V. Zaitsev. Necessary and sufficient conditions of stability in the large. рнн, Ф. 46, ЧЩР. 5, 1982 З. (in Russian).

16.      V. G. Veretennikov, V.N. Seregin. Investigation of oscillations of nonlinear systems. MAI, Moscow, 1982 (in Russian).

17.      V. G. Veretennikov. Investigation of oscillations of gyro horizon. In the collection of contributions "The investigation of periodic motions and stability of mechanical systems", MAI, Moscow, 1983 (in Russian).

18.      V. G. Veretennikov.љ Stability and oscillations of nonlinear systems. "Nauka", Moscow, 1984 (in Russian).

19.      V. G. Veretennikov, V.V. Zaitsev. The use of Lyapunov's second method to estimate regions of stability and attraction. PMM, v. 48, issue 5, 1984 (in Russian).

20.      V. G. Veretennikov, V.V. Zaitsev.љ Lyapunov's second method for the investigation of stability in large. In monography "The stability of motion", "Nauka", Siberian branch of AS USSR, 1985 (in Russian).

21.      V. G. Veretennikov, V.V. Zaitsev. љLyapunov's second method. The estimations of domains for stability and attraction. MAI, Moscow, 1986 (in Russian).

22.      V. G. Veretennikov, I.A. Korolev. Investigations of the oscillations for essentially nonlinear systems with internal resonance. PMM, v.51, issue 4, 1987 (in Russian).

23.      V. G. Veretennikov, I.I. Karpov. Theoretical mechanics. Conclusion and analyses of equations with the help of computer. "Vishaja shkola", Moscow, 1990 (in Russian).

24.      V. G. Veretennikov. On the algorithm of construction of periodic solutions for Lyapunov's systems. "Applied mechanics", v. 27, ї 2, 1991, Kiev (in Russian).

25.      V. G. Veretennikov. The use of normal form in Kamenkov's method of construction of periodic solutions. In the collection of contributions "Some problems of dynamic of mechanical systems", MAI, Moscow, 1991 (in Russian).

26.      V. G. Veretennikov, V.A. Sinitsyn. Conclusion the equations of motions for systems with variable structure. "Bulletin of MAI", v. 1, issue 2, 1994, Moscow (in Russian).

27.      V. G. Veretennikov, V.A. Sinitsyn. On the stability of motion for the systems with internal non elastic friction. In the collection of contributions "Problems of mechanics for controlled motions", Perm, State University of Perm, 1994 (in Russian).

28.      V.G.Veretennikov, V.A. Sinitsyn.љ Dynamics of sheaf between rigid body and material points. In the collection of contributions "Actual problems of classical and celestial mechanics", "Elf", Moscow, 1998 (in Russian).

29.      V. G. Veretennikov, I.I. Karpov, J.G. Markov. Oscillation processes in mechanical system with elastic and dissipative elements. MAI, Moscow, 1998 (in Russian).

30.      V. G. Veretennikov, V.A. Sinitsyn. Dynamics of great body in the atmosphere of planets. "Bulletin of MAI", v. 6, issue 1, 1999, Moscow (in Russian).

31.      V.G.Veretennikov, V.A. Sinitsyn.љ Investigation of integral principle of mechanics. In collections of scientific and methodical contributions "Theoretical mechanics", issue 23, Moscow state university, 2000, Moscow (in Russian).

32.      V. G. Veretennikov, V.A. Sinitsyn. Method of variable action. "Fismatlit", 2002, Moscow

33.      V. G. Veretennikov, V.A. Sinitsyn. Analysis of dynamics of wheel-deformable rail system. News RAS "Mechanics of rigid body", ї 2, 2002, Moscow (in Russian).

34.      V. G. Veretennikov, V.A. Sinitsyn.љ Dynamics of point with variable mass. J. "Actual problems of aviation and space systems (processes, models, experiment)", v.8, ї 2(16), 2003, Kazan (in Russian).

35.      V. G. Veretennikov, V.A. Sinitsyn.љ Integral principle of equality of action to counteraction. In collections of scientific and methodical contributions "Theoretical mechanics", issue 24, Moscow state university, 2003, Moscow (in Russian).

36.      V. G. Veretennikov, V.A. Sinitsyn.љ Mathematical modelling of motion of solar sail in space operating. J. "Actual problems of aviation and space systems (processes, models, experiment)", ї 1(17), 2004, Kazan (in Russian).

37.      V. G. Veretennikov, V.A. Sinitsyn. On the concept of "natural system" and homogeneous property of lagrangians. In collections of scientific and methodical contributions "Theoretical mechanics", issue 25, Moscow state university, 2004, Moscow (in Russian).

38.      V. G. Veretennikov, V.A. Sinitsyn. Analytical principles of investigations of dynamics. Bulletin of MAI, v.12, ї 2, 2005, Moscow (in Russian).

39.      V. G. Veretennikov, V.A. Sinitsyn. Class of dynamical systems with constraints .RAS, v. 404, ї 5, 2005 (in Russian).

40.      V. G. Veretennikov, V.A. Sinitsyn. Principle of the predicativity and "cambridge's problem" of motion for circuit. RAS, v.402, ї 1, 2005 (in Russian).

41.      V. G. Veretennikov, V.A. Sinitsyn.љ Theoretical mechanics. Additions to common sections. "Fizmatlit", 2006, Moscow (in Russian).

42.      V. G. Veretennikov, V.A. Sinitsyn. On the theory of Chaplygin's reducing factor. љRAS, v.412, ї 5, 2007 (in Russian).

43.      V. G. Veretennikov, V.A. Sinitsyn. Development of Rayleigh's method with the help of principle of changeable action. RAS, v.418, ї 6, 2008 (in Russian).