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.