Second Order Riemannian Mechanics

Abstract

Abstract The technique of covariant differentiation is being implemented into the single integral variational calculus with second derivatives in Riemannian manifolds and into the Grässer-Rund-Weyssenhoff homogeneous generalized canonical framework for the parameter-invariant variational problem. As an example of physical nature we relate to each other the equations of motion with higher derivatives that describe the physical phenomena known as ‘Zitterbewegung’, the electromagnetic self-interaction, and spherical top. Although two of these phenomena, the quiver (the ‘Zitterbewegung’) and the radiation friction, have been known only within the framework of Special Relativity, our approach allows to generalize the corresponding equations of motion to the curved spacetime. From the geometrical point of view these generalizations depend neither on the signature of the Riemannian space nor on its dimension.