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.