  With the discovery during the past decade of a large number of extrasolar
planets orbiting their parent stars at a distance lower than 0.1 astronomical
unit (and the launch and the preparation of dedicated space missions such as
CoRoT and KEPLER), with the position of inner natural satellites around giant
planets in our Solar System and with the existence of very closed but separated
binary stars, tidal interaction has to be carefully studied. In particular, a
question arises about the validity of usual approximations used in the
modelling of this interaction. The purpose of this paper is to examine the step
beyond the ponctual approximation for the tidal perturber. To achieve this aim,
the gravitational interaction between two extended bodies and more precisely
the interaction between mass multipole moments of their gravitational fields
and the associated tidal phenomena are studied. Use of Cartesian Symmetric
Trace Free (STF) tensors, of their relation with spherical harmonics and of the
Kaula's transform enables to derive analytically the tidal and mutual
interaction potentials as well as the associated disturbing functions in
extended bodies systems. The tidal and mutual interaction potentials of two
extended bodies are derived. Next, the external gravitational potential of such
tidally disturbed extended body is obtained, using the Love's number theory, as
well as the associated disturbing function. Finally, the dynamical evolution
equations for such systems are given in their more general form without any
linearization. The dynamical equations for the gravitational and tidal
interactions between extended bodies and associated dynamics are derived in a
form where they could be directly implemented to perform coherent numerical
simulations of planetary systems or multiple stars tidal evolution.
