The relativistic models of two-body systems with the
     retarded interactions: qualitative aspects and
          theorems about the degrees of freedom
                            
                            
 
                        V.I.Zhdanov
       Astronomical Observatory of Kiev University
      Observatorna St., 3;   Kiev 254053   Ukraine
         E-mail: AOKU@GLUK.APC.ORG (To: V.Zhdanov)
 
The  relativistic equations of motion  must  take  into
account   the   finite   speed   of   propagation    of
interactions.  As the result, the relativistic  systems
have  infinite numbers of degrees of freedom,  and  the
motion of interacting bodies in any relativistic  field
theory  cannot  be  described  (in  general  case)   by
ordinary  differential  equations.  Nevertheless,   the
approximations  known in the General Relativity  reduce
the  dynamics to the instantaneous form with the finite
number  of  the initial value parameters.  The  correct
relativistic theory of motion must explain the role  of
the  infinite number of degrees of freedom, so that  we
must  be sure about the approximation methods and  that
there  is  no  loss of physical solutions. This  report
presents  a summary of exact results of the  author  on
these problems.
      Because there is no explicit equations of  motion
is  General  Relativity, we confine  ourselves  to  the
exact  Poincare - invariant models of two-body systems.
The evolution equations can be reduced to a functional-
differential system of a neutral type with  respect  to
the   trajectories  of  the  bodies.   We   study   the
qualitative  behavior and present  the  conditions  for
existence  and  uniqueness of the  global  trajectories
specified   by  instantaneous  initial  positions   and
velocities.  It  is  shown  that  under  some   general
assumptions   it   is   possible  to   characterize   a
relativistic  N-body  system  by  a  finite  number  of
degrees of freedom. The convergence of iteration method
is  proved  yielding instantaneous equations of  motion
for    regular    weakly   relativistic   trajectories.
Specificity  of  the  equations of  motion  in  General
Relativity in view of the above problems is discussed.