  We study the dynamical interactions of mass systems in equilibrium under
their own gravity that mutually exert and experience gravitational forces. The
method we employ is to model the dynamical evolution of two isolated bars,
hosted within the same galactic system, under their mutual gravitational
interaction. In this study we present an analytical treatment of the secular
evolution of two bars that oscillate with respect one another. Two cases of
interaction, with and without geometrical deformation, are discussed. In the
latter case, the bars are described as modified Jacobi ellipsoids. These
triaxial systems are formed by a rotating fluid mass in gravitational
equilibrium with its own rotational velocity and the gravitational field of the
other bar. The governing equation for the variation of their relative angular
separation is then numerically integrated, which also provides the time
evolution of the geometrical parameters of the bodies. The case of rigid,
non-deformable, bars produces in some cases an oscillatory motion in the bodies
similar to that of a harmonic oscillator. For the other case, a deformable
rotating body that can be represented by a modified Jacobi ellipsoid under the
influence of an exterior massive body will change its rotational velocity to
escape from the attracting body, just as if the gravitational torque exerted by
the exterior body were of opposite sign. Instead, the exchange of angular
momentum will cause the Jacobian body to modify its geometry by enlarging its
long axis, located in the plane of rotation, thus decreasing its axial ratios.
