  We study the superconducting current of a Josephson junction (JJ) coupled to
an external nanomagnet driven by a time dependent magnetic field both without
and in the presence of an external AC drive. We provide an analytic, albeit
perturbative, solution for the Landau-Lifshitz (LL) equations governing the
coupled JJ-nanomagnet system in the presence of a magnetic field with arbitrary
time-dependence oriented along the easy axis of the nanomagnet's magnetization
and in the limit of weak dimensionless coupling $\epsilon_0$ between the JJ and
the nanomagnet. We show the existence of Shapiro-like steps in the I-V
characteristics of the JJ subjected to a voltage bias for a constant or
periodically varying magnetic field and explore the effect of rotation of the
magnetic field and the presence of an external AC drive on these steps. We
support our analytic results with exact numerical solution of the LL equations.
We also extend our results to dissipative nanomagnets by providing a
perturbative solution to the Landau-Lifshitz-Gilbert (LLG) equations for weak
dissipation. We study the fate of magnetization-induced Shapiro steps in the
presence of dissipation both from our analytical results and via numerical
solution of the coupled LLG equations. We discuss experiments which can test
our theory.
