Noncyclic geometric phase and its non-Abelian generalization

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