The Dynamics of Parallel Transport

There is a primitive relationship between Riemannian geometry and classical mechanics: by Newton’s first law, free particles move along straight lines. Consequently, geodesics can be scrutinized by experimenting with the paths of free particles. This brings dynamical concepts to Riemannian geometry, and a recent text in the subject ([3]) also presents classical mechanics, even though its author’s aims seem purely geometric. Indeed, as Abraham and Marsden ([1], p. 225) mention, “. . . one can start with the Hamiltonian approach as basic and define the covariant derivative, and so forth, in terms of this structure”. Some aspects of Riemannian geometry are often explained from the point of view what ‘residents’ of the manifold would naturally experience. Classical mechanics gives precise meaning to these explanations insofar as they are about geodesics: residents of a Riemannian manifold would first learn the geometry of their world by understanding the ballistics of its point-like particles. Imagine you live in the geometry defined by a metric g on a Riemannian manifold Q, and you want to transport a vector v along a geodesic qðtÞ, 0 t a. For instance you may need to establish reference vectors at various locations for the purpose of covariant differentiation. A natural procedure would be obtain a small, rigid, one-dimensional filament. You place the center of the filament at qð0Þ and align the filament with v. Tossing the filament without spinning it so its center has initial velocity q0ð0Þ, you hope that the center will continue along the geodesic qðtÞ. The result of transporting v along q is the direction of the filament when at time t 1⁄4 a its center arrives at qðaÞ. In this article I show that this produces parallel transport in Q if the Riemann curvature of the metric g is isotropic. I do this by showing that parallel transport solves the Euler-Lagrange equations for a Lagrangian system that describes the motion of small filaments. As an aside I obtain a new coordinate-independent formula (eq. (13)) for the Levi-Civita connection. If the curvature is anisotropic then filaments do not generally parallel transport because their direction couples through the Riemann curvature to their position and velocity. I avoid this by replacing the filaments with small spheres. This leads to a Lagrangian system on the orthonormal frame bundle Qn of Q, and to another characterization of the Levi-Civita connection: the Levi-Civita connection on Qn is the zero level set of the Noether momentum (with respect to the Lagrangian) of the right-hand action of SOðnÞ on Qn. As an application consider the Foucault pendulum, which is simply a bob at the surface of the Earth suspended by a wire and excited without local angular momentum and which can be observed over a 24 hour period. This system is a standard example of the consequences of fictitious forces in a noninertial frame and is common in elementary mechanics texts. As observation shows, and these textbooks derive, the plane of the motion slowly turns about the vertical axis as though it were being parallel transported along its circle of latitude. This is immediate from my link between dynamics and parallel transport, as follows. Since the period of the pendulum is fast compared with period of the Earth’s rotation, we can view the pendulum as a rigid filament being translated on the surface of the spherical Earth – this is an averaging approximation. By this article the motion of such a filament, viewed from an inertial frame, is by parallel translation. While many textbooks derive this venerable result, and note the coincidence of the motion of the Foucault pendulum and parallel translation, to my knowledge the explanation of it through the linking of dynamics with parallel transport is new. The Foucault pendulum is also discussed in [6], where the angle from an initial orientation of the pendulum to its orientation 24 hours later is shown to be the holonomy of the Hannay-Berry connection.