  We investigate the weak excitations of a system made up of two condensates
trapped in a Bose-Hubbard ring and coupled by an interspecies repulsive
interaction. Our approach, based on the Bogoliubov approximation scheme, shows
that one can reduce the problem Hamiltonian to the sum of sub-Hamiltonians
$\hat{H}_k$, each one associated to momentum modes $\pm k$. Each $\hat{H}_k$ is
then recognized to be an element of a dynamical algebra. This uncommon and
remarkable property allows us to present a straightforward diagonalization
scheme, to find constants of motion, to highlight the significant microscopic
processes, and to compute their time evolution. The proposed solution scheme is
applied to a simple but still very interesting closed circuit, the trimer. The
dynamics of low-energy excitations, corresponding to weakly-populated vortices,
is investigated considering different choices of the initial conditions, and
the angular-momentum transfer between the two condensates is evidenced.
Finally, the condition for which the spectral collapse and dynamical
instability are observed is derived analytically.
