Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.

A dynamic stiffness-based framework for harmonic input estimation and response reconstruction considering damage

In structural health monitoring(SHM),the measurement is point-wise but structures are continuous.Thus,input estimation has become a hot research subject with which the full-field structural response can be calculated with a finite element model(FEM).This paper proposes a framework based on the dynamic stiffness theory,to estimate harmonic input,reconstruct responses,and to localize damages from seriously deficient measurements.To begin,Fourier transform converts the dynamic equilibrium equation to an equivalent static one in the frequency domain,which is underdetermined since the dimension of measurement vector is far less than the FEM-node number.The principal component analysis has been adopted to“compress”the under-determined equation,and formed an over-determined equation to estimate the unknown input.Then,inverse Fourier transform converts the estimated input in the frequency domain to the time domain.Applying this to the FEM can reconstruct the target responses.If a structure is damaged,the estimated nodal force can localize the damage.To improve the damage-detection accuracy,a multi-measurement-based indicator has been proposed.Numerical simulations have validated that the proposed framework can capably estimate input and reconstruct multi-types of full-field responses,and the damage indicator can localize minor damages even with the existence of noise.