anholonomy

Anholonomy is the property of a system to fail to return to its initial state after a closed loop, due to intrinsic curvature, a nontrivial connection, or nonintegrable constraints. In differential geometry, it is most precisely associated with frames and parallel transport: a frame is holonomic if it can be derived from a coordinate basis; if not, the frame is anholonomic. The noncommutativity of the frame fields, expressed by [e_i, e_j] ≠ 0, is captured by structure functions called anholonomy coefficients. Such frames are common in curved spaces and in settings with noncoordinate bases.

When a vector or tensor is parallel transported around a closed loop, the result may differ from the starting state. The net effect is called holonomy; when this effect is nontrivial due to curvature or a nontrivial connection, the situation is described as anholonomy. A classic geometric illustration is parallel transport on a sphere: transporting a vector around a loop yields a rotated orientation related to the loop’s area and the sphere’s curvature, a manifestation of anholonomic behavior.

Anholonomy appears in several domains. In physics, the geometric or Berry phase is an example of anholonomy in parameter space, where a system acquires a phase after cyclic evolution. In classical mechanics, nonholonomic constraints prevent expressing certain velocity constraints as simple position constraints, a related idea in which the allowed motions do not arise from a single coordinate system. In mathematics, the concept is tied to the curvature of a connection and the holonomy group, which encodes all possible anholonomic effects from parallel transport around loops.