  This report is a review of Darwin's classical theory of bodily tides in which
we present the analytical expressions for the orbital and rotational evolution
of the bodies and for the energy dissipation rates due to their tidal
interaction. General formulas are given which do not depend on any assumption
linking the tidal lags to the frequencies of the corresponding tidal waves
(except that equal frequency harmonics are assumed to span equal lags).
Emphasis is given to the cases of companions having reached one of the two
possible final states: (1) the super-synchronous stationary rotation resulting
from the vanishing of the average tidal torque; (2) the capture into a 1:1
spin-orbit resonance (true synchronization). In these cases, the energy
dissipation is controlled by the tidal harmonic with period equal to the
orbital period (instead of the semi-diurnal tide) and the singularity due to
the vanishing of the geometric phase lag does not exist. It is also shown that
the true synchronization with non-zero eccentricity is only possible if an
extra torque exists opposite to the tidal torque. The theory is developed
assuming that this additional torque is produced by an equatorial permanent
asymmetry in the companion. The results are model-dependent and the theory is
developed only to the second degree in eccentricity and inclination
(obliquity). It can easily be extended to higher orders, but formal accuracy
will not be a real improvement as long as the physics of the processes leading
to tidal lags is not better known.
