  A clear separation of the time scales governing the dynamics of "slow" and
"fast" degrees of freedom often serves as a prerequisite for the emergence of
an independent low-energy theory. Here, we consider (slow) classical spins
exchange coupled to a tight-binding system of (fast) conduction electrons. The
effective equations of motion are derived under the constraint that the quantum
state of the electron system at any instant of time $t$ lies in the
$n$-dimensional low-energy subspace for the corresponding spin configuration at
$t$. The effective low-energy theory unfolds itself straightforwardly and takes
the form of a non-abelian gauge theory with the gauge freedom given by the
arbitrariness of the basis spanning the instantaneous low-energy sector. The
holonomic constraint generates a gauge covariant spin-Berry curvature tensor in
the equations of motion for the classical spins. In the non-abelian theory for
$n>1$, opposed to the $n=1$ adiabatic spin dynamics theory, the spin-Berry
curvature is generically nonzero, even for time-reversal symmetric systems. Its
expectation value with the representation of the electron state is gauge
invariant and gives rise to an additional {\em geometrical} spin torque.
Besides anomalous precession, the $n\ge 2$ theory also captures the spin
nutational motion, which is usually considered as a retardation effect. This is
demonstrated by proof-of-principle numerical calculations for a minimal model
with a single classical spin. Already for $n=2$ and in parameter regimes where
the $n=1$ adiabatic theory breaks down, we find good agreement with results
obtained from the full (unconstrained) theory.
