  We study the effects of the coupling between magnetization dynamics and the
electronic degrees of freedom in a heterostructure of a metallic nanomagnet
with dynamic magnetization coupled with a superconductor containing a steady
spin-splitting field. We predict how this system exhibits a non-linear spin
torque, which can be driven either with a temperature difference or a voltage
across the interface. We generalize this notion to arbitrary magnetization
precession by deriving a Keldysh action for the interface, describing the
coupled charge, heat and spin transport in the presence of a precessing
magnetization. We characterize the effect of superconductivity on the
precession damping and the anti-damping torques. We also predict the full
non-linear characteristic of the Onsager counterparts of the torque, showing up
via pumped charge and heat currents. For the latter, we predict a spin-pumping
cooling effect, where the magnetization dynamics can cool either the nanomagnet
or the superconductor.
