Zilch and simultaneous partial differential equations (Q2534914)

Maxwell's equations in free space yield a conservation equation relating energy density and energy flux, the electric and magnetic field terms entering these quantities not being differentiated. In recent years, further quantities have been discovered satisfying conservation equations, these quantities involving the first derivatives of the electric and magnetic fields, but not derivable from the energy density and the energy flux by differentiation. This paper generalizes these discoveries to \(n\) simultaneous equations of the form \(\operatorname{curl}\mathfrak w_i= \sum a_{ij}\partial \mathfrak w_j/\partial t\). Generalized reciprocity flux density in the particular vector product form \(\sum x_{ij} \mathfrak x_i\wedge \mathfrak y_j\) is sought, the condition for its existence being the possibility of finding solutions to the matrix equation \(XA +A'X =0\), \(A\) denoting the system matrix, \(X\) the coefficients occurring in the reciprocity flux density, and a prime the transpose. The solution of this matrix equation is sought in terms of the Jordan canonical form of the matrix \(A\). The existence of matrices \(X\) is shown to imply that the set of characteristic roots of \(A\) should contain at least one pair \(\pm q\). The total number of independent matrices \(X\) that exists is expressed in terms of the entries in the Segre characteristic of the matrix \(A\). Generalized energy and zilch forms are defined by a suitable choice of the vectors \(\mathfrak x_i\) and \(\mathfrak y_j\), the paper concluding with an examination of the consequences of source terms in the original equations, leading to rates of loss of energy and zilch per unit volume.