The purpose of this Appendix is to elaborate on the invariant dynamics on the first two energy shells. This is a special case of the exact dynamics of the ground state modes and the nth energy shell discussed in Subsection 16.2.1 with n = 2. For this purpose let us recall the following tables:

Eigenfunctions corresponding to ? ? ₁= ?2( n=1)

Eigenfunctions corresponding to ? ? ₂= ?6( n=2)

It is easy to see, utilizing (16.6)

This implies, since x ² = 0.5( x ² ? y ²) + 0.5(1 ? z ²) , that

This further implies, thanks to the rotation symmetry of the sphere and the eigenfunctions (which are homogeneous polynomials of degree 1 and 2),

Thus for

we have

We observe

Hence the dynamics of the coefficients are given by, for the simple case of no topography ( h = 0), no damping ( = 0) and no external forcing ( = 0)

This system of linear equations can be written in a compact form

where the matrix A is given by

In general the above matrix A is periodic in t with period ?/ ? (since a _{1 1}, b ₁₁ are periodic in t with period ?/ ?) and Floquet theory is needed to solve such system.

Special Case 1: a ₁₁ = a