  The Landau-Lifshitz equation describes the time-evolution of magnetic
dipoles, and can be derived by taking the classical limit of a quantum
mechanical spin Hamiltonian. To take this limit, one constrains the many-body
quantum state to a tensor product of coherent states, thereby neglecting
entanglement between sites. Expectation values of the quantum spin operators
produce the usual classical spin dipoles. One may also consider expectation
values of polynomials of the spin operators, leading to quadrupole and
higher-order spin moments, which satisfy a dynamical equation of motion that
generalizes the Landau-Lifshitz dynamics [Zhang and Batista, Phys. Rev. B 104,
104409 (2021)]. Here, we reformulate the dynamics of these $N^2-1$ generalized
spin components as a mean-field Schr\"odinger equation on the $N$-dimensional
coherent state. This viewpoint suggests efficient integration methods that
respect the local symplectic structure of the classical spin dynamics.
