  We perform a symmetry analysis of the zero-temperature instabilities of the
tetragonal phase of Ca3Mn2O7 and Ca3Ti2O7 which is stable at high temperature.
We introduce order parameters to characterize each of the possible lattice
distortions in order to construct a Landau free energy which elucidates the
proposed group-subgroup relations for structural transitions in these systems.
We include the coupling between the unstable distortion modes and the
macroscopic strain tensor. We also analyze the symmetry of the dominantly
antiferromagnetic ordering which allows weak ferromagnetism. We show that in
this phase the weak ferromagnetic moment and the spontaneous ferroelectric
polarization are coupled, so that rotating one of these ordering by applying an
external electric or magnetic field one can rotate the other ordering. We
discuss the number of different domains (including phase domains) which exist
in each of the phases and indicate how these may be observed.
