  L$1_0$ FePt is a technologically important material for a range of novel data
storage applications. In the ordered FePt structure the normally non-magnetic
Pt ion acquires a magnetic moment, which depends on the local field originating
from the neighboring Fe atoms. In this work a model of FePt is constructed,
where the induced Pt moment is simulated by using combined longitudinal and
rotational spin dynamics. The model is parameterized to include a linear
variation of the moment with the exchange field, so that at the Pt site the
magnetic moment depends on the Fe ordering. The Curie temperature of FePt is
calculated and agrees well with similar models that incorporate the Pt dynamics
through an effective Fe-only Hamiltonian. By computing the dynamic correlation
function the anisotropy field and the Gilbert damping are extracted over a
range of temperatures. The anisotropy exhibits a power-law dependence with
temperature with exponent $n\approx2.1$. This agrees well with what observed
experimentally and it is obtained without including a two-ion anisotropy term
as in other approaches. Our work shows that incorporating longitudinal
fluctuations into spin dynamics calculations is crucial for understanding the
properties of materials with induced moments.
