Group theoretical analysis of a rotating shallow liquid in a rigid container

A Lie group analysis is undertaken for a nonlinear system which models the motion of a rotating shallow liquid in a rigid basin. The Lie algebra of the symmetry group is presented for elliptic and circular paraboloidal basins. In the elliptic case, the symmetry algebra is a six-dimensional real Lie algebra. In the circular case, the symmetry algebra is nine dimensional. Finite group transformations are constructed which, in the circular paraboloidal case, deliver a theorem concerning the time evolution of a key moment of inertia during the motion. In the elliptic paraboloidal case, a result concerning the motion of the centre of gravity of the liquid is retrieved. The investigation ends with symmetry reduction of the original system and the generation of group-invariant solutions which correspond to various initial data.