  Spectral properties of the Hamiltonian function which characterizes a trapped
ion are investigated. In order to study semiclassical dynamics of trapped ions,
coherent state orbits are introduced as sub-manifolds of the quantum state
space, with the K\"ahler structure induced by the transition probability. The
time dependent variational principle (TDVP) is applied on coherent states'
orbits. The Hamilton equations of motion on K\"ahler manifolds of the type of
classical phase spaces naturally arise. The associated classical Hamiltonian is
obtained from the expected values on symplectic coherent states of the quantum
Hamiltonian. Spectral information is thus coded within the phase portrait. We
deal with the bosonic realization of the Lie algebra of the SU(1,1) group,
which we particularize for the case of an ion confined in a combined, Paul and
Penning trap. This formalism can be applied to Hamiltonians which are nonlinear
in the infinitesimal generators of a dynamical symmetry group, such as the case
of ions confined in electrodynamic traps. Discrete quasienergy spectra are
obtained and the corresponding quasienergy states are explicitly realized as
coherent states parameterized by the stable solutions of the corresponding
classical equations of motion. A correspondence between quantum and classical
stability domains is thus established, using the Husimi representation.
