  Continuing work initiated in an earlier publication [H. Asada, Phys. Rev. D
{\bf 80}, 064021 (2009)], the gravitational radiation reaction to Lagrange's
equilateral triangular solution of the three-body problem is investigated in an
analytic method. The previous work is based on the energy balance argument,
which is sufficient for a two-body system because the number of degrees of
freedom (the semimajor axis and the eccentricity in quasi-Keplerian cases, for
instance) equals that of the constants of motion such as the total energy and
the orbital angular momentum. In a system with three (or more) bodies, however,
the number of degrees of freedom is more than that of the constants of motion.
Therefore, the present paper discusses the evolution of the triangular system
by directly treating the gravitational radiation reaction force to each body.
The perturbed equations of motion are solved by using the Laplace transform
technique. It is found that the triangular configuration is adiabatically
shrinking and is kept in equilibrium by increasing the orbital frequency due to
the radiation reaction if the mass ratios satisfy the Newtonian stability
condition. Long-term stability involving the first post-Newtonian corrections
is also discussed.
