By Dominic G. B. Edelen

This new sequence Mechanics and Physics of Discrete structures goals to supply a coherent photo of the trendy improvement of discrete actual structures. every one quantity will provide an orderly point of view of disciplines corresponding to molecular dynamics, crystal mechanics and/or physics, dislocation, and so on. emphasised specifically are the basics of mechanics and physics that play an important function in engineering purposes. quantity 1, Gauge concept and Defects in Solids, provides an in depth improvement of a rational conception of the dynamics of defects and harm in solids. recommendations to box equations are used to figure out stresses, dislocation densities and currents that come up from histories of loading of obstacles of our bodies. Analysed intimately is a gauge thought with a gauge team that's not semi-simple, and whose motion happens on the classical macroscopic point. Yang-Mills thought is utilized the place the nation variables are elastic displacements in solids, choice of mechanical and electromagnetic observables by way of selection of gauge stipulations is verified, and practices of classical dislocation idea are derived from first rules

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1 Sect. 4] Elasticity via the Exterior Calculus 35 It has the realization x* = Q)xXX , A T) + k , t=T l + k. 22) T h e action of the group G x t thus picks the whole history of the material body u p and moves it rigidly to a new history in E x R and t h e n translates the whole history along the time axis. All is now in readiness for the variational arguments. Let Ψ d e n o t e the stored energy per unit mass of the material body on £ x i?. Noting that the Lagrangian function for a mechanical system is the kinetic energy minus the potential energy, we have 0 4 3 3 L = p { ^ V - ^ } .

44) and hence we must have d d ^ = 0. 45) i A will be such that f = άχ' only if 0 = d f = \{d F A -d F ) = 0, dF B A dX A B A dX B + {d V - d F )dX l A A 4 A A dT ; that is, e dF ABC l B c l 4 A = d V. 46) W h e n these kinematic compatability conditions are satisfied by the F ' s and the V s , the equations άχ =/' can be solved on any star-shaped region by use of the h o m o t o p y o p e r a t o r H , ι Sect. 47) w h e r e the /c's are the values of the ^ ' s at the center. 46). 32)). Because R is star-shaped with respect t o any of its points as center, the Poincare l e m m a holds in R .

So that the A"s are the same before as after G acts. This is an essential aspect of classical gauge theory. T h e group G acts only as an internal symmetry group; it does not change the underlying manifold on which the state variables are defined. It is for precisely this reason that the material description will be used in the gauge theory of defects, for in that case, the action integral is an integral over the space R of reference configuration histories, while the internal symmetry group acts on the material space Af x R.