"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.
                  
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