Noncyclic geometric phase and its non-Abelian generalization

We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin-1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) field.