Gravity Waves on the Surface of the Sphere

We propose a Hamiltonian model for gravity waves on the surface of a uid layer surrounding a gravitating sphere. The general equations of motion are non-local and can be used as a starting point for simpler model equations, which can be derived systematically by expanding the Hamiltonian functional. In this paper, we study in detail the small wave amplitude regime. The rst order non-linear terms can be eliminated by a formal canonical transformation. Similarly, many of the second order terms can be eliminated. The resulting model has the feature that it leaves invariant several nite dimensional subspaces on which the motion is integrable.