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