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In this paper, the solution to long standing problem of deriving Maxwell&rsquo, s equations and Lorentz force from first principles, i.e., from Coulomb&rsquo, s law, is presented. This problem was studied by many authors throughout history but it was never satisfactorily solved, and it was never solved for charges in arbitrary motion. In this paper, relativistically correct Li&eacute, nard&ndash, Wiechert potentials for charges in arbitrary motion and Maxwell equations are both derived directly from Coulomb&rsquo, s law by careful mathematical analysis of the moment just before the charge in motion stops. In the second part of this paper, the electrodynamic energy conservation principle is derived directly from Coulomb&rsquo, s law by using similar approach. From this energy conservation principle the Lorentz force is derived. To make these derivations possible, the generalized Helmholtz theorem was derived along with two novel vector identities. The special relativity was not used in our derivations, and the results show that electromagnetism as a whole is not the consequence of special relativity, but it is rather the consequence of time retardation.

In a previous work [Campisi, M. (2005). On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem. Studies in History and Philosophy of Modern Physics, 36, 275–290] we have addressed the mechanical foundations of equilibrium thermodynamics on the basis of the generalized Helmholtz theorem. It was found that the volume entropy provides a good mechanical analogue of thermodynamic entropy because it satisfies the heat theorem and it is an adiabatic invariant. This property explains the “equal” sign in Clausius principle ( S f ⩾ S i ) in a purely mechanical way and suggests that the volume entropy might explain the “larger than” sign (i.e. the law of entropy increase) if non-adiabatic transformations are considered. Based on the principles of microscopic (quantum or classical) mechanics we prove here that, provided the initial equilibrium satisfies the natural condition of decreasing ordering of probabilities, the expectation value of the volume entropy cannot decrease for arbitrary transformations performed by some external source of work on an insulated system. This can be regarded as a rigorous quantum-mechanical proof of the second law. We discuss how this result relates to the minimal work principle and how it improves on previous attempts. The natural evolution of entropy is towards larger values because the natural state of matter is at positive temperature. Actually the law of entropy decrease holds in artificially prepared negative temperature systems.

There are some limitations on the statement of classic Helmholtz theorem although it has broad applications. Actually, it only applies to simply connected domain with single boundary surface and does not provide any conclusion about the domain where discontinuities of field function exist. However, discontinuity is often encountered in practice, for example, the location of surface sources or interface of two kinds of medium. Meanwhile, most existing versions of Helmholtz theorem are imprecise and imperfect to some extent. This paper not only tries to present a precise statement and rigorous proof on classic Helmholtz theorem with the accuracy of mathematical language and logical strictness, but also generalizes it to the case of multiply connected domain and obtains a generalized Helmholtz theorem in the sense of Lebesgue measure and Lebesgue integral defined on three-dimensional Euclidean space. Meanwhile, our proof and reasoning are more sufficient and perfect. As an important application of the generalized Helmholtz theorem, the concepts of irrotational and solenoidal vector function are emphasized. The generalized Helmholtz theorem and the present conclusion should have important reference value in disciplines including vector analysis such as electromagnetics.

This paper develops a generalized Helmholtz theorem for an arbitrary localized time-varying vector function F(r,t) and shows that the mathematical relations commonly referred to as Maxwell’s equations can be derived from that theorem alone.

An elegant but seldom appreciated effort to provide a mechanical model of equilibrium thermodynamics dates back to the Helmholtz theorem (HT). According to this theorem, the thermodynamic relations hold mechanically (without probabilistic assumptions) in the case of one-dimensional monocyclic systems. Thanks to a discrete picture of the phase space, Boltzmann was able to apply the HT to multi-dimensional ergodic systems, suggesting that the thermodynamic relations we observe in macroscopic systems at equilibrium are a direct consequence of the microscopic laws of dynamics alone. Here I review Boltzmann’s argument and show that, using the language of the modern ergodic theory, it can be safely re-expressed on a continuumphase space as a generalized Helmholtz theorem(GHT), which can be readily proved. Along the way the agreement between the Helmholtz–Boltzmann theory and that of P. Hertz (based on adiabatic invariance) is revealed. Both theories, in fact, lead to define the entropy as the logarithmof the phase-space volum e enclosed by the constant energy hypersurface (volume entropy). r 2005 Elsevier Ltd. All rights reserved.

In a recent paper Davis formulated a generalized Helmholtz theorem for a time-varying vector field in terms of the Lorenz gauge retarded potentials. The purposes of this comment are to point out that Davis's generalization of the theorem is a version of the extension of the Helmholtz theorem formulated some years ago by McQuistan and also by Jefimenko and more recently by the present author and to show that Davis's expression for the time-dependent vector field is also valid for potentials in gauges other than the Lorenz gaug

The widely used Helmholtz theorem separates an arbitrary vector function into two parts: one is an irrotational component which can be expressed by a scalar function and the other is a rotational part which can only be expressed by another vector function. So it is considered as an incomplete projection of a vector function. In this paper, the generalized theorem is proved. In this theorem an arbitrary vector function is separated into three parts; each can be expressed as a scalar function. It is the complete projection for a vector function in vector wave function space.

By making use of the Stokes operator and its corresponding dual operator, which are introduced here, a short proof of the Helmholtz theorem for antisymmetric second‐rank tensor fields in Minkowski space‐time is presented.

The generalization of the classical Helmholtz theorem for time-dependent vector fields is presented. The application to physics shows that the group of gauge transformations may be much larger than that commonly used.

The classical theory of Helmholtz is extended to the case of one- and two-forms in the space of two, three and four dimensions. We show that the Helmholtz decomposition into the scalar, vector and tensor potential is invariant with respect to a transformation which is a generalization of the classical Abelian gauge transformation. Applications of the theorem to the ideal compressible fluids, classical electrodynamics and continuum dislocation theory indicate that there exists a possibility of participation of the constitutive equations in the decomposition of multi-forms.

This paper deals with the theory of dielectric waveguides in a graded‐index media. The theory is first developed in terms of electrical and magnetic potentials and a gauge condition which is a generalization of the Lorentz gauge used in classical electromagnetic theory for the isotropic, homogeneous medium. The existence of these potentials under the generalized Lorentz gauge in the context of a generalized Helmholtz theorem are then proven. Even though this generalized gauge uncouples the governing equations, it does not lead to partial differential equations in the normal form. To alleviate this problem, superpotentials of the Hertzian variety are introduced and it is shown how equations governing the vector superpotential can be reduced to a form which is amenable to standard analysis. Then several uniqueness theorems associated with this superpotential are proven. As an illustration of the use of the superpotential and its relative merits, it is applied to certain waveguide problems in an inhomogeneous medium with variable permittivity.