Line defect evolution in finite-dimensional manifolds

A fiber bundle Y having as a prototype fiber a finite-dimensional, differentiable manifold, not embedded in any linear space, and as a base a space-time tube constructed on the n-dimensional point space is the geometric environment considered here. After assigning a Lagrangian to its first jet bundle, the notion of defect and the representation of its evolution in this setting is discussed first. Then the geometry is restricted to a base involving fit regions of a three-dimensional Euclidean space, and the balances of actions on defects in the abstract fiber which can be pictured on the basis of Y by a smooth space-type line l are derived in a non-purely conservative setting (a d'Alember-Lagrange principle is involved). The main result is related to the case involving non-constant peculiar energy along l: the covariance of the action balances along l, including the configurational ones, is related to the validity of an integral Nöther-like relation which has the meaning of action power relative to the defect motion.