  The problem studied is formulated in the title: proper mass variation under
gravitational and Coulomb force action in Relativistic Mechanics of point
particle. The novelty is that equations of motion are obtained in the
relativistic Lagrangean framework for conservative force fields under
assumption of field dependent proper mass. The dependence of proper mass on
field strength is derived from the equations of particle motion. The result is
the elimination of a classical 1/r divergence. It is shown that a photon in a
gravitational field may be described in terms of a refracting massless medium.
This makes the gravity phenomenon compatible with SR Dynamic framework. New
results concerning gravitational properties of particle and photon, as well as
an experimental test of predicted deviation from 1/r^2 classical Coulomb force
law are discussed. The conclusion is made that the approach of field-dependent
proper mass is perspective for better understanding GR problems and further
studies on divergence-free field theory development.
  Key words: Relativity; gravity; Coulomb; particle; photon; speed of light;
proper mass variation.
  PACS: 03.30.+p, 04.20.-g
