American Institute of Mathematical Sciences

March  2012, 17(2): 575-596. doi: 10.3934/dcdsb.2012.17.575

Line defect evolution in finite-dimensional manifolds

 1 DICeA, University of Florence, via Santa Marta 3, I-50139 Firenze, Italy

Received  October 2011 Published  December 2011

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'Alembert-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.
Citation: Paolo Maria Mariano. Line defect evolution in finite-dimensional manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 575-596. doi: 10.3934/dcdsb.2012.17.575
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