The energy exchange between the degrees of freedom of a multi-dimensional oscillator is a widespread phenomenon. A new two-dimensional formalism has been derived to extend the one-dimensional results of transverse coherent instabilities in the presence of x-y coupling. It shows that coupling between the transverse planes of
a particle beam leads to a sharing of the oscillation energy, including the growth rates of coherent instabilities. Furthermore, in the presence of a frequency spread, there can also be a partition of Landau damping for 'optimum' coupling. Measurements performed in the CERN-PS for the LHC beam and applications to other machines
will be discussed.