On the variational calculus and corona fields: charge drift, duality and degeneracy

xThis paper considers the fundamentals of the variational approach to the c harge-drift equations as part of the generalized field theory of electrical discharges in gases. It is in furtherance of the published theory on charg ed gaseous flows in which some variational results have been given based on continuity principles. These results were developed as part of the Lagrang ian approach to charge drift, which is the equivalent of the more commonly used Eulerian continuity equation. A complete and exhaustive set of optimiz ing principles, on which finite-element methods can be based, would be of c onsiderable help to any problem solver using Lagrangian methods; and it is with the global search for such expressions that this work is concerned. Th e author has previously succeeded in showing that solving the governing equ ations of charge drift is equivalent to finding minimizing electric potenti als for certain integrals in which the charge distributions are known. The dual of these principles, in which optimal charge distributions are sought for given potentials, was not given in earlier work as an appropriate integ ral could not be found. This situation is investigated, and proofs are give n of the non-existence (degeneracy) of these dual principles for both singl e and multiple ionic flows. The work is put into the general context of the search for solutions of ionic flow problems in a gaseous medium and is of particular relevance to corona discharges and their applications, aerial io nization phenomena and gas-insulated systems. It is of decisive importance in revealing the existence or non-existence of full finite-element solution s to charge-drift problems, as it clearly reveals when this method can give results and when not. It further indicates how the finite-element approach can be used jointly with other numerical methods. The paper has other fund amental implications, for it shows whether or not variational proofs can be constructed of major results in the theory of charge drift which have alre ady been derived in the literature by other means.