  In the spirit of multi-scale modeling, we develop a theoretical framework for
spin-lattice coupling that connects, on the one hand, to ab initio calculations
of spin-lattice coupling parameters and, on the other hand, to the
magneto-elastic continuum theory. The derived Hamiltonian describes a closed
system of spin and lattice degrees of freedom and explicitly conserves the
total momentum, angular momentum and energy. Using a new numerical
implementation that corrects earlier Suzuki-Trotter decompositions we perform
simulations on the basis of the resulting equations of motion to investigate
the combined magnetic and mechanical motion of a ferromagnetic nanoparticle,
thereby validating our developed method. In addition to the ferromagnetic
resonance mode of the spin system we find another low-frequency mechanical
response and a rotation of the particle according to the Einstein-de-Haas
effect. The framework developed herein will enable the use of multi-scale
modeling for investigating and understanding a broad range of
magneto-mechanical phenomena from slow to ultrafast time scales.
