  The isoscalar giant monopole resonance (ISGMR) and isovector giant dipole
resonance (IVGDR) in finite nuclei are studied in the framework of a
relativistic transport approach. The kinetic equations are derived within an
effective nucleon-meson field theory in the Relativistic Mean Field (RMF)
scheme, even extended to density dependent vertices. Small amplitude
oscillations are analysed using the Relativistic Vlasov (RV) approach, i.e.
neglecting nucleon collision terms. The time evolution of the isoscalar
monopole moment and isovector dipole moment and the corresponding Fourier power
spectra are discussed. In the case of ^{208}Pb we study in detail the
dependence of the monopole response on the effective mass and symmetry energy
at saturation given by the used covariant effective interaction. We show that a
reduced m^* and a larger a_4 can compensate the effect on the ISGMR energy
centroid of a much larger compressibility modulus K_{nm}. This result is
important in order to overcome the conflicting determination of the nuclear
compressibility between non-relativistic and relativistic effective
interactions. For the symmetry energy dynamical effects, we carefully analyze
the influence of the inclusion of an effective isovector scalar channel,
  \delta-meson field, with constant and density dependent couplings. We show
the relevance of the $slope$ (or pressure) of the symmetry energy at saturation
on the ISGMR and IVGDR modes for neutron-rich systems. Density dependent
vertices are not much affecting our conclusions. Following as a guidance some
extended dispersion relations in nuclear matter, we see two main reasons for
that, the smoothness of the density dependences around saturation and the
presence of compensation effects coming from rearrangement terms.

Nuclear Theory