Higher-Order Fermi–Walker Transport Dynamics and the Induced Geometric Phase

Abstract

We define higher-order Fermi–Walker derivatives of vector fields along curves in Euclidean 3-space using the Frenet–Serret frame. From the incompatibility of successive Fermi–Walker transports, we derive a geometric phase—termed the Fermi–Walker flow transport phase—which depends on the intrinsic geometry of the curve. We then examine this phase for various curve evolutions, including binormal, complex modified Korteweg–de Vries, and other integrable motions. Our formulation provides an explicit and unified method for computing the induced geometric phases in these settings.