New Interpretation of Newton's Law of Universal Gravitation
Journal of High Energy Physics Gravitation and Cosmology
Elliptical motions of orbital bodies are treated here using Fourier series, Fortescue sequence components and Clarke's instantaneous space vectors, quantities largely employed on electrical power systems analyses. Using this methodology, which evidences the analogy between orbital systems and autonomous second-order electrical systems, a new theory is presented in this article, in which it is demonstrated that Newton's gravitational fields can also be treated as a composition of Hook's elastic
... of Hook's elastic type fields, using the superposition principle. In fact, there is an identity between the equations of both laws. Furthermore, an energy analysis is conducted, and new concepts of power are introduced, which can help a better understanding of the physical mechanism of these quantities on both mechanical and electrical systems. The author believes that, as a practical consequence, elastic type gravitational fields can be artificially produced with modern engineering technologies, leading to possible satellites navigation techniques, with less dependency of external sources of energy and, even, new forms of energy sources for general purposes. This reinterpretation of orbital mechanics may also be complementary to conventional study, with implications for other theories such as relativistic, quantum, string theory and others. Journal of High Energy Physics, Gravitation and Cosmology tive, negative and the zero-sequence symmetrical circuits. The correspondent electrical quantities are called symmetrical components, respectively, positive, negative and zero sequence components, whose respective phasors are of equal amplitudes and symmetrically spaced from each other, on a complex plane; excepted the zero-sequence phasors which are coincident. On dealing with unbalanced and non-sinusoidal circuits, Fourier analysis is applied to each phase of the polyphase systems and, then, sets of symmetrical sequence circuits are obtained for each harmonic order, using the same method above described. More recently, for three-phase circuits, a transformation, called Clarke's transformation , transforms those circuits on equivalent two-phase circuits, which brings all the information of the original circuits, i.e., unbalanced or balanced and/or sinusoidal or non-sinusoidal electrical quantities, such as voltages, currents, fluxes and charges. Clarke's transformation leads to a unique rotating vector, the instantaneous space vector (ISV)     , on a complex plane. After applying Clarke's transformation, then Fourier analysis is performed, and the positive complex Fourier coefficients, related to positive harmonic orders, are separated from the negative ones, using rotating unity vectors that rotate anti-clockwise and clockwise, respectively. This is an alternative way to obtain the Fortescue sequence components, since they are, in fact, the Fourier coefficients themselves. Fortescue's method is performed on the frequency domain. Clarke's, in turn, is performed on the time domain. In this work, Fourier analysis is applied to the study of elliptical orbital motions, which are represented on a complex plane, as compositions of two related orthogonal motions. In this way, all the mechanical quantities are treated as ISVs, and Clarke's inverse transformation can be performed to obtain the equivalent mechanical three-phase systems (other inverse transformations to polyphase systems are possible). In doing so, the analogy between orbital mechanical systems and three-phase electrical systems is evidenced. More precisely speaking, the dynamical behavior of elliptical orbital motions is analogous to the behavior of autonomous, second order, loss-less polyphase electrical circuits, with only reactive elements, operating under unbalanced and non-sinusoidal conditions. The figures generated for the mechanical variables, on the complex plane, are all Lissajous figures of elliptical type. Fourier analyses for orbital studies are not new, however, here, emphases are given to the geometric and dynamic properties of such motions, using the above-mentioned transformations of Fortescue and Clarke. Furthermore, the author develops a careful analysis on how the stored energy is distributed among the harmonic motions and their flux between the positive and negative sequence inside each harmonic motion, and among motions of different harmonic orders. Those energy fluxes can be better studied on future works, when the intention is Journal of High Energy Physics, Gravitation and Cosmology how to manipulate them, when studying the possibility of new economical energy storage and consumption. As a major contribution of this work, it is demonstrated that Newton's universal law for gravitational fields, the inverse square law, can be studied using Hook's law for elastic objects. The author believes that new horizons are opened to the study and control of orbital dynamics, with possible applications of electric power systems control techniques, with low-loss reactive electromechanical elements. Finally, the used concepts can also be extended to electrostatic forces and to the astronomy realms, and beyond the Newtonian mechanics.