"The History of Computer Algebra Application in Mechanics in Russia (Soviet Union)" G.B.Efimov, M.V.Grosheva Keldysh Institute of Applied Mathematics RAS Institute of Mechanics Moscow State Lomonosov University Moskow, 125047, Miusskaya sq.4, Russia efimov@applmat.msk.su grosheva@inmech.msu.su Present Review considers the History of Mechanical Applications of Symbolical Algebraic Manipulations (SAM) or Computer Algebra (CA) in our country - for almost forty years. >From the very beginning mechanicians was among the leaders of SAM application - due to vast problems, important for them (spacecraft dynamics, theory of strength), as well as due to presence and readiness of experts, capable to develop mathematical methods and to apply/develop computer programs for SAM. The first attempts of computer symbolical manipulations in Soviet Unions were made more than thirty years ago. In the work of Nobel laureate L.V.KANTORORVICH and his followers the problems of SAM and their applications were considered, in particular in the theory of elasticity. N.N.YANENKO and his colleagues solved the problems of differential schemes theory. D.E.OKHOTSIMSKY, V.A.BRUMBERG and others applied SAM in celestial mechanics. M.CHUBAROV investigated a stability problem in linear systems. Despite the computers had only the few K of memory at that time, some interesting problems were solved successively with them. In the middle of 1960's and in the beginning of 1970's the old groups of workers were expanded and new ones have arisen in the area of SAM. The groups were working in Moscow, Leningrad, Novosibirsk, Gorky, Tomsk, Kazan, Harkov, Irkutsk, Vilnius.The areas of researches were: stability in mechanics, oscillation and control theory, celestial mechanics and elasticity, group theory in fluid mechanics, mathematics, physics and computorial problems. In Kiev V.A.GLUSHKOV with his colleagues have created the computer "MIR-2", the first realization of the symbolic- numerical language ANALYTIK by hardware means. Many mathematical and mechanical problems were solved and many non-trivial algorithms realized on these computers. Later ANALYTIK was realized both by hardware on Hulliet-Packers and IBM machines and by software. SAM systems were created with both high and low level languages. As an interesting example of high level language we can mention V.F.TURCHIN with his original language Refal. S.S.LAVROV and his colleagues worked on Lisp. A.P.ERSHOV was initiator of creation the universal tipe SAM. From the other hand, applied specialists, physicists and mechanicians, often created specialized SAM program systems for solving their problems, and sometimes preferred Algol or Fortran. Numerous applications had very large range: from nuclear physics and General Relativity, multi-bodies dynamics and stability to differential geometry and analysis. In the end of the 70-ies SAM were used more often and have obtained rather wide popularity. Conferences in Novosibirsk, Irkutsk, in Moscow University contributed to this popularity. At the known conferences in Dubna the foreign systems (REDUCE, for example) were demonstrated and widely distributed. REDUCE algorithms were created in mechanics applications domain as well as in physics one. The large event was the great conference on SAM in mechanics in 1984 in Gorky. The majority of experts in SAM from our country were participating it, and achievements of works for 20 years were demonstrated. There were such known scientists among participants as A.A.SAMARSKY, N.N.GOVORUN, D.M.KLIMOV, V.M.MATROSOV, D.V.SHIRKOV and others. The programmers have demonstrated the SAM systems, physicists, mechanicians and mathematicians presented various problems and algorithms. In theory of elasticity, mechanics of fluid and celestial mechanics SAM were used for construction of solutions in series. Formal derivation of equations of movement, their averaging, analysis of stability, generation of numerical programs for simulation - were applied for various difficult mechanical systems. The large difference schemes, formulas of estimation of accuracy were formed. The group properties of systems of equations, their integrals were investigated. The SAM applications in multi bodies dynamics (more then 15 groups from Moscow, S-Petersburg, Irkutsk, Perm, Vilnius etc.) were analyzed in special conferences and work-shops in S-Petersburg, Vilnius, Irkutsk and Moscow University. At present SAM became recognized and customary tool for solution of problems of mechanics of multi-body systems, celestial mechanics, cavitation, strength, physics, theory of control. Now many domestic systems, including specialized, have finished the work. Their role was taken by the programs in systems REDUCE, MAPLE, MATHEMATYCA and other ones. But the experts on SAM and their applications in various areas continue to work. The experience of work, solution of specific SAM problems is present. SAM are applied in new areas (chemistry, verification of programs etc.), but SAM conventional applications in mechanics as always are popular. SAM are widely used in mathematical education from the very beginning: in formation system programmers on SAM, for training for SAM-systems, for mechanics and physics education. In many Universities the courses on SAM, their systems, algorithms and applications are read. The students are taught to work with SAM, to construct and realize symbolical algorithms, to apply SAM in mechanics, mathematics, physics, to participate in development of systems. The geography of SAM application in education has very large domain: from S.Petersburg to Irkutsk, >from Perm and Dubna to Kiev and Lvov. Information about many systems and mathematical and mechanical algorithms in them were assembled by authors of this report. The modern data are collecting in information bank on SAM: "Computer Symbolic manipulations in problems of mechanics" (Institute of Mechanics. Moscow State Lomonosov University. Michurynski pr.1, Moscow 117192, Russia. M.V.Grosheva: tel.(095) 939-5478,e-mail: grosheva@inmech.msu.su). The work was carried out under support of RFFI grant N 96-15 -97229, N 98-01-00941 and Russian Universities, Mathematical Simulation. [1] Grosheva M.V., Efimov G.B. Systems for analytical calculation at computer. - In: Packages of applied programs. Analytical transformations. Moscow, Nauka, 1988, 5-30. (In Russian). [2] Grosheva M.V., Samsonov V.A. Information bank: Computer Symbolic manipulations in problems of mechanics. - In: Computer Analytical manipulations in automatic of scientific research. Vilnius, 1990, p.92. (In Russian). [3] Efimov G.B., Grosheva M.V. On History of use the Computerized Symbolical Manipulation's in Mechanics. - Preprint of Inst.of Mech. Moscow St. Lomonosov University, No 16, Moscow. 1995. 44 pp. (In Russian). [5] Miola A.M., Pottosin I.V. A bibliography of soviet works in algebraic manipulations. - SIGSAM Bull., 1981, 15, N 1, 5-7. [6] Brumberg V.A., Tarasevich S.V., Vasiliev N.N. Specialized celestial mechanics systems for symbolic manipulations.- Celest. Mech. 1989, 45, 149-162.