A New Generic Class of Beltrami “Force-Free” Fields. Part-I: Theoretical considerations
نویسندگان
چکیده
We report on a new general class of solutions of the Beltrami equation, with special characteristics. We also provide examples of solutions that also satisfy Maxwell equations. A subset of these solutions can be isolated which corresponds to " gauge " fields. A special projective geometry of vacuum fields is also revealed and discussed. 1. Introduction The notion of a force-free field comes directly from the work of 19 th century mathematician Eugenio Beltrami (1835-1899) in hydrodynamics [1], [2]. In fact Beltrami made a very important contribution by a direct comparison between electrodynamics and hydrodynamics probably inspired by Lord Kelvin's vortex model of the atom. The central notion is that of a generalized eigenvalue of a rotation operator. As this is allowed to vary also with space and time it is preferable to call it the eigen-vorticity and we use this term for the rest of this paper. The special case of constant eigen-vorticity is often met in the literature under the name of a Trkalian flow from the Czech physicist and mathematician Viktor Trkal (1888-1955) who has done similar work independently back at 1919 [3]. The subject is shown to be of significance in fluid mechanics, electromagnetism, magnetohydrodynamics and astrophysics while there is a proposition that the famous Vacuum Energy may be composed from such force-free fields [2]. In the following, we briefly introduce the subject of Beltrami fields and their general significance in section 2. In section 3, we present a general theorem for the construction of a class of solutions of the Beltrami equation. In section 4, we discuss the problem of the independent existence of such electric and magnetic fields either in vacuum or through appropriate sources and boundary conditions. In section 5, we further analyze the consequences of the previous abstract observations in the effort to develop a concrete algorithmic condition for constructing similar solutions of Maxwell equations. In section 6, we conclude on the possibility of a new class of devices that may lead to new applications in engineering electromagnetics as we intend to show in the second part of this article. This also provides strong indications that a new area of high energy physics is possible with macroscopic instruments effecting truly macroscopic results.
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