Geometric phases in classical three-body dynamics

In the classical dynamics of the $N$-body problem, a geometric phase is meaningful both physically and differential geometrically. Physically, a geometric phase corresponds to a net rotation due to nonzero internal angular momentum (with respect to a moving frame) contributing to the total angular momentum of a system. Differential geometrically, a geometric phase corresponds to a net rotation described by the holonomy of a connection. This result is demonstrated for the dynamics of a moving frame for three-body systems in terms of various coordinates, such as polar Jacobi coordinates, Guichardet coordinates, and the mutual distances between the bodies. In classical molecular dynamics,\footnote{F. J. Lin, {\it Discrete and Continuous Dynamical Systems Supplement} {\bf 2007}, 655 - 666 (2007).} geometric phases have been physically observed both experimentally and computationally. For example, a geometric phase has been observed experimentally in the rotation of the recoil angle of a departing atom in a dissociating triatomic molecule (a three-body system) due to rotation of the remaining diatomic fragment, and a geometric phase has been observed computationally in the overall rotation of a flexible protein molecule (an $N$-body system) due to its internal motions.