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The dynamics of a set of identical spins interacting with another one througha time-dependent coupling gives rise to a gyroscopic equation with a variableLarmor frequency and, more importantly, with an operator playing the role aLarmor vector. The subsequent technical complexity is due to non-trivialalgebraic relations between multiple inner products coming from thenon-commutative algebra of the angular momenta. A general formalism is derivedgiving the integrated solution valid for all values of the involved spins, andseveral applications of the formalism are treated in details. Among otherresults, it is seen that, starting from a fully polarised state for the set ofidentical spins, their total spin can at most only partially flip (in themean); this somewhat surprising fact means that the memory of the initial stateis kept for ever but varying the coupling constant allows to adjust at will thepossible polarisation of the final state. The robustness of the initial stateis shown to depend on the nature fermionic or bosonic of the perturbing spinand also on the size of the collection of identical spins.

A method for solving gyroscopic equations with operators