  The Hamilton-Lagrange action principle for Relativistic Schr\"odinger Theory
(RST) is converted to a variational principle (with constraints) for the
stationary bound states. The groundstate energy is the minimally possible value
of the corresponding energy functional and the relativistic energy eigenvalue
equations do appear as the corresponding variational equations. The matter part
of these eigenvalue equations is a relativistic generalization of the
well-known Ritz principle in non-relativistic quantum mechanics which however
disregards the dynamical character of the particle interactions. If the latter
are included in the proposed principle of minimal energy for the bound states,
one obtains a closed dynamical system for both matter and gauge fields. The new
variational principle enables the development of variational techniques for
solving approximately the energy eigenvalue equations. As a demonstration, the
positronium groundstate is treated in great detail. Here a simple exponential
trial function is sufficient in order to reproduce the (exact) result of
conventional quantum mechanics where the relativistic and spin effects are
neglected.
