  We calculate the change of the properties of a resonator, when coupled to a
semiclassical spin by means of the magnetic field. Starting with the Lagrangian
of the complete system, we provide an analytical expression for the linear
response function for the motion of the resonator, thereby considering the
influence of the resonator on the spin and vice versa. This analysis shows that
the resonance frequency and effective dissipation factor can change
significantly due to the relaxation times of the spin. We first derive this for
a system consisting of a spin and mechanical resonator and thereafter apply the
same calculations to an electromagnetic resonator. Moreover, the applicability
of the method is generalized to a resonator coupled to general two and more
level systems, providing a key to understand some of the problems of two level
systems in quantum devices.
