Though the vanishing of the covariant divergence does not directly imply a conservation law, there is some meaning both to conservation and to non-conservation of energy in General Relativity. In this paper the method of slow changes previously applied to elucidate the conservation of energy is used to study changes in cylindrical systems with rotation. It is shown that in such systems only a single quantity is conserved that can be identified with angular momentum. It corresponds to the symmetry imposed, and agrees with the Killing vector based analysis of J. Winicour. On the other hand, as previously shown, axial energy transport prevents the existence of a conservation theorem for mass. This conserved angular momentum is intrinsic to the system in that its value is unaffected by the state of rotation of the coordinate system used to define it. Machian space drag is also examined. It emerges that angular momentum and space drag behave very differently as thicker and thicker spinning cylinders are studied.