Vladimir Savchin
Doctor of Physics and Mathematics
Professor,
s.M. Nikol’skii Mathematical Institute
Think it through!
1981
Graduated with honors from the Peoples’ Friendship University named after Patrice Lumumba in the specialty of «Mathematics». He was awarded the qualification of «Mathematician. Lecturer of mathematics in higher and secondary learning establishments». He was also awarded the qualification of «Translator from French into Russian».
1981-1984
Post-graduate student at the P. Lumumba Peoples’ Friendship University.
1984
Candidate thesis “Method of Ostrogradskii and inverse problems of mechanics” was defended.
1984-1987
Staff member of the computing center of the Peoples’ Friendship University, senior lecturer.
1987-1988
Scientific trip to the University of California, Santa Barbara (USA).
1988-1993
Senior lecturer, associate professor in the Department of Mathematics and Informatics at the P. Lumumba Peoples’ Friendship University.
1992
Doctoral thesis “Mathematical methods of mechanics of infinite-dimensional non-potential systems” for the degree of Doctor of Physics and Mathematics was defended at the M.V. Lomonosov Moscow State University.
1993-2005
Professor in the Department of Mathematical Analysis.
1995
Academic title Professor was awarded.
2005-2018
Professor in the Department of Mathematical Analysis and Theory of Functions.
2018 - present time
Professor of the S.M. Nikol’skii Mathematical Institute.
2010
Gratitude of the Federal Agency for Education.
2013
The breastplate and honorary title “Honorary worker of higher professional education of the Russian Federation”.
2017
Honorary badge “Veteran of the RUDN”.
Teaching
- V.M. Savchin worked out a number of new special courses for master's students, in particular:
- “Variational methods for the study of operators” (direction “Functional Methods in Differential Equations and Interdisciplinary Research”),
- “Inverse problems of the calculus of variations” (direction “Functional Methods in Differential Equations and Interdisciplinary Studies").
- Courses at the RUDN University:
- “Mathematical analysis” (direction “Applied Mathematics and Informatics”) for bachelor’s students,
- “Inverse problems of the calculus of variations” (direction “Functional Methods in Differential Equations and Interdisciplinary Research”) for bachelor’s students.
Science
Author of monographs:
- Mathematical methods of mechanics of infinite-dimensional systems. M.: UDN, 1991, 237 p.,
- Variational principles for nonpotential operators //Results of science and technology. Contemporary Problems of Mathematics. Latest Achievement. M.: VINITI, 1992.Vol. 40. p. 3-178 (co-authors: Filippov V. M., Shorokhov S. G.) (translated and published in the USA).
- Research is closely linked with the works by G. Helmholtz, G. Birkhoff et al. on variational principles for finite-dimensional systems. The problem of the extension and development of their results to the case of infinite-dimensional systems was of theoretical and practical interest. It is interrelated with the problems of modern Hamiltonian mechanics, the existence of solutions to inverse problems of the calculus of variations for given equations and such algebraic structures as Lie algebras and Lie-admissible algebras. Major works are devoted to development of mathematical methods of mechanics of infinite-dimensional systems, development of methods for constructing variational formulations of the equations of motion of infinite-dimensional nonpotential systems and their application to such specific problems as the representation of evolution equations in the form of Hamilton's equations, the introduction of Poisson brackets and symmetric brackets in Eulerian and non-Eulerian classes of functionals, the search of algebraic structures associated with the equations of motion, finding symmetries and first integrals of evolution equations.
V.M.Savchin:
- Obtained a general criterion for the B-potentiality of operators with respect to local bilinear forms.
- Developed constructive methods of determination of integral variational principles for wide classes of equations of motion of infinite-dimensional nonpotential systems using Eulerian and non-Eulerian classes of functionals.
- Established the connection of equations of motion of nonpotential systems with different Poisson brackets and Lie-admissible algebras and on this basis extended the scope of application of classical Hamilton's mechanics.
- Obtained the operator equation - an analogue of ordinary Birkhoff’s differential equations - and established its significance in the mechanics of infinite-dimensional systems. Found the necessary and sufficient conditions for the representation of the given evolutionary equations in the form of Birkhhoff’s operator equation and obtained the formulas for the construction of the corresponding operators.
- Extended the methods of investigation of canonical equations of the rank greater than zero to infinite-dimensional nonpotential systems.
- For the first time ever formulated the inverse problem of the calculus of variations for a very general system of partial differential difference equations and obtained the conditions of its potentiality.
- Worked out an operator approach providing a unique methodology for the investigation of a number of properties of the equations of motion of both finite dimensional and infinite-dimensional systems.
- Theoretical results are applied, in particular, to the system of Navier-Stokes equations describing the flow of liquids, to the Korteweg-de-Vries equation describing nonlinear waves, in studies of a number of dissipative systems.
Scientific interests
- variational principles for nonpotential operators,
- inverse problems of variational calculus,
- symmetries,
- the relationship of dynamical systems with algebraic structures.
One presents numerous approaches for the construction of variational principles for equations with operators which, in general, are nonpotential. One considers separately linear and nonlinear ordinary differential equations, partial and integropartial differential equations. One constructs and investigates both extremal and stationary variational principles and one gives applications of these principles in theoretical physics and in analytic mechanics. A series of unsolved problems are indicated. The survey is intended for mathematicians, physicists, working in both theoretical and applied areas, as well as for graduate students of physics and mathematics.
There is suggested the notion of Вu-potentiality of operator N with
respect to local bilinear form, and necessary and sufficient conditions of such
generalized potentiality are obtained. Formulas for the construction of the
corresponding integral variational action are given. Theoretical results are
illustrated by two examples.
In the terms of neсessary and sufficient condition there is established the structure of operators of the given evolution equation admitting the direct variational formulation with respect to the fixed bilinear form and the symmetric operator d/dt.
There is described a quite general structure of Lie-admissible algebra in a space of Gâteaux differentiable operators. It is a natural generalization of Lie-bracкets.
There is constructed a semibounded functional whose minimum is attained on the solutions of the boundary value problem for the nonlinear unsteady Navier - Stokes equations.
Using methods of nonlinear functional analysis there is established the structure of the given type of evolution operator equation admitting the direct variational formulation.
Using methods of nonlinear functional analysis, we define the structure of an evolution operator equation of second order that can be formulated in direct variational terms.
The existence of direct variational formulations for a wide class of second order evolutionary equations is investigated.
The problem of existence of variational principles for wide classes of generally nonlinear differential-difference equations with nonpotential operators is investigated.
We establish a connection between symmetries of functionals and symmetries of the corresponding Euler–Lagrange equations. A similar problem is investigated for equations with quasi-B u-potential operators.
The relationship between the symmetries of the functionals and the corresponding Euler–Lagrange equations is established. A similar question is also investigated in the case of equation.
The necessary and sufficient conditions for the representation of the operator equation with the first time derivative in the form of a Hamilton-permissible equation are obtained.
The operator equation with the second time derivative is presented in the form of a Hamilton-valid equation. The relationship between the solutions of the Hamilton-admissible equations and the corresponding Hamilton equations is established.
The use of variational methods for constructing sufficiently accurate approximate solutions to this system requires the existence of a corresponding variational principle-the solution of inverse problems of the calculus of variations. Within the framework of Euler functionals, variational principles may not exist. But if we extend the class of functionals, it can allow us to obtain variational formulations of these problems. Naturally, the problem arises of constructively defining the corresponding functionals - non-classical Hamilton actions-and applying them to find approximate solutions to given boundary value problems. The main goal of the paper is to present a scheme for constructing indirect variational formulations for given evolutionary problems and to demonstrate the effective use of the non-classical Hamilton action for constructing approximate solutions with high accuracy for a given dissipative problem.
Using the methods of nonlinear analysis, the connection between the first integrals and the absolute integral invariants of some evolutionary equations is established, similar to the case of the dynamics of finite-dimensional systems.
General structures of admissible Lie algebras in spaces of operators differentiable by Gato are introduced and their connection with the symmetries of operator equations and the mechanics of infinite-dimensional systems is established.
The main goal of the paper is to present a scheme for constructing indirect variational formulations for given evolutionary problems and to demonstrate the effective use of non-classical Hamiltonian actions for constructing approximate solutions with high accuracy for a given dissipative problem. The paper uses the concepts and methods of nonlinear functional analysis and modern calculus of variations.
S. L. Sobolev's works on low-amplitude oscillations of a rotating fluid in the 1940s aroused great interest. After their publication, I. G. Petrovsky stressed the importance of studying general differential equations and systems that are not resolved with respect to the higher-order time derivative. In this regard, it is natural to study the question of the existence of their variational formulations. It can be considered as an inverse problem of the calculus of variations. The main purpose of this paper is to study this problem for the Sobolev system. The key object is the criterion of potentiality. On this basis, we prove the non-potentiality of the operator of the boundary value problem for a system of Sobolev partial differential equations with respect to the classical bilinear form. It is shown that this system does not allow a matrix variational factor of a given form. Thus, the equations of the Sobolev system cannot be derived from the classical Hamilton principle. The question is raised whether there is a functional that is semi-bounded for solving this boundary value problem. An algorithm for the constructive determination of such a functional is proposed. The main advantage of the constructed functional-action is the possibility of using direct variational methods.
The main purpose of this work is to study the potentiality of a discrete system obtained from the system of the form C(t,u)u˙(t)+E(t,u)=0 with continuous time. The definition of potentiality of the corresponding discrete system is introduced. Necessary and sufficient conditions for its potentiality with respect to a given bilinear form are obtained. The algorithm for the construction of the corresponding functional—the analogue of the Hamiltonian action—is presented. The illustrative example is given.
By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator S to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples.
The main purpose of this work is, first, a construction of the indirect Hamilton's variational principle for the problem of motion of a pendulum with a vibration suspension with friction, oscillating along a straight line making a small angle with the vertical line. Second, the construction on its basis of the difference scheme. Third, to carry out its investigation by methods of numerical analysis. Methods. The problem of motion of the indicated pendulum is considering as a particular case of the given boundary problem for a nonlinear second order differential equations. For the solution of problem of its variational formulation there is used the criterion of potentiality of operators - the symmetry of the Gateaux derivative of nonlinear operator of the given problem. This criterion is also used for the construction of variational multiplier and the corresponding Hamilton's variational principle. On its basis there is constructed and investigated a discrete analog of the given boundary problem and a problem of motion of the pendulum. Results. It is proved that the operator of the given boundary problem is not potential with respect to the classical bilinear form. There is found a variational multiplier and constructed the corresponding indirect Hamilton's variational principle. On its basis there is obtained a discrete analog of the given boundary problem and its solution is found. As particular cases one can deduce from that the corresponding results for the problem of motion of the pendulum. There are performed numerical experiments, establishing the dependence of solutions of the problem of motion of the pendulum on the change of parameters. Conclusion. There is worked out a variational approach to the construction of two difference schemes for the problem of a pendulum with a suspension with friction, oscillating along a straight line making a small angle with the vertical line. There are presented results of numerical simulation under different parameters of the problem. Numerical results show that under sufficiently small amplitude and sufficiently big frequency of the oscillations of the point of suspension the pendulum realizes a periodical motion.