New Interpretation of Newton s Law of Universal Gravitation

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1 Journal of Hig Energy Pysics, Gravitation and Cosmology, 07, 3, ttp:// ISS Online: ISS Print: ew Interpretation of ewton s Law of Universal Gravitation Dalgerti L. Milanese Electrical Engineering Department, State University Júlio de Mesquita Filo UESP, Ila Solteira, Brazil How to cite tis paper: Milanese, D.L. (07) ew Interpretation of ewton s Law of Universal Gravitation. Journal of Hig Energy Pysics, Gravitation and Cosmology, 3, ttps://doi.org/0.436/jepgc Received: June 5, 07 Accepted: September, 07 Publised: September 6, 07 Copyrigt 07 by autors and Scientific Researc Publising Inc. Tis work is licensed under te Creative Commons Attribution International License (CC BY 4.0). ttp://creativecommons.org/licenses/by/4.0/ Open Access Abstract Elliptical motions of orbital bodies are treated ere using Fourier series, Fortescue sequence components and Clarke s instantaneous space vectors, quantities largely employed on electrical power systems analyses. Using tis metodology, wic evidences te analogy between orbital systems and autonomous second-order electrical systems, a new teory is presented in tis article, in wic it is demonstrated tat ewton s gravitational fields can also be treated as a composition of Hook s elastic type fields, using te superposition principle. In fact, tere is an identity between te equations of bot laws. Furtermore, an energy analysis is conducted, and new concepts of power are introduced, wic can elp a better understanding of te pysical mecanism of tese quantities on bot mecanical and electrical systems. Te autor believes tat, as a practical consequence, elastic type gravitational fields can be artificially produced wit modern engineering tecnologies, leading to possible satellites navigation tecniques, wit less dependency of external sources of energy and, even, new forms of energy sources for general purposes. Tis reinterpretation of orbital mecanics may also be complementary to conventional study, wit implications for oter teories suc as relativistic, quantum, string teory and oters. Keywords Orbital Motions, Gravity, ewton s Second Law, Hook s Elastic Forces, Fourier, Electrical Circuits, Transformations of Fortescue and Clarke, Instantaneous Space Vectors, Instantaneous Complex Power. Introduction On steady-state analyses of polypase electrical circuits, operating under unbalanced and sinusoidal conditions, usually, electrical engineers use Fortescue s metod [] to decompose tose circuits on sets of balanced circuits, called posi- DOI: 0.436/jepgc Sep. 6, Journal of Hig Energy Pysics, Gravitation and Cosmology

2 tive, negative and te zero-sequence symmetrical circuits. Te correspondent electrical quantities are called symmetrical components, respectively, positive, negative and zero sequence components, wose respective pasors are of equal amplitudes and symmetrically spaced from eac oter, on a complex plane; excepted te zero-sequence pasors wic are coincident. On dealing wit unbalanced and non-sinusoidal circuits, Fourier analysis is applied to eac pase of te polypase systems and, ten, sets of symmetrical sequence circuits are obtained for eac armonic order, using te same metod above described. More recently, for tree-pase circuits, a transformation, called Clarke s transformation [], transforms tose circuits on equivalent two-pase circuits, wic brings all te information of te original circuits, i.e., unbalanced or balanced and/or sinusoidal or non-sinusoidal electrical quantities, suc as voltages, currents, fluxes and carges. Clarke s transformation leads to a unique rotating vector, te instantaneous space vector (ISV) [] [3] [4] [5] [6], on a complex plane. After applying Clarke s transformation, ten Fourier analysis is performed, and te positive complex Fourier coefficients, related to positive armonic orders, are separated from te negative ones, using rotating unity vectors tat rotate anti-clockwise and clockwise, respectively. Tis is an alternative way to obtain te Fortescue sequence components, since tey are, in fact, te Fourier coefficients temselves. Fortescue s metod is performed on te frequency domain. Clarke s, in turn, is performed on te time domain. In tis work, Fourier analysis is applied to te study of elliptical orbital motions, wic are represented on a complex plane, as compositions of two related ortogonal motions. In tis way, all te mecanical quantities are treated as ISVs, and Clarke s inverse transformation can be performed to obtain te equivalent mecanical tree-pase systems (oter inverse transformations to polypase systems are possible). In doing so, te analogy between orbital mecanical systems and tree-pase electrical systems is evidenced. More precisely speaking, te dynamical beavior of elliptical orbital motions is analogous to te beavior of autonomous, second order, loss-less polypase electrical circuits, wit only reactive elements, operating under unbalanced and non-sinusoidal conditions. Te figures generated for te mecanical variables, on te complex plane, are all Lissajous figures of elliptical type. Fourier analyses for orbital studies are not new, owever, ere, empases are given to te geometric and dynamic properties of suc motions, using te above-mentioned transformations of Fortescue and Clarke. Furtermore, te autor develops a careful analysis on ow te stored energy is distributed among te armonic motions and teir flux between te positive and negative sequence inside eac armonic motion, and among motions of different armonic orders. Tose energy fluxes can be better studied on future works, wen te intention is DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

3 ow to manipulate tem, wen studying te possibility of new economical energy storage and consumption. As a major contribution of tis work, it is demonstrated tat ewton s universal law for gravitational fields, te inverse square law, can be studied using Hook s law for elastic objects. Te autor believes tat new orizons are opened to te study and control of orbital dynamics, wit possible applications of electric power systems control tecniques, wit low-loss reactive electromecanical elements. Finally, te used concepts can also be extended to electrostatic forces and to te astronomy realms, and beyond te ewtonian mecanics.. Discrete Fourier Series of te Position, Velocity and Force Vectors of Elliptical Orbital Motions, in Terms of Sequence Components For orbital motions, Kepler s equations are used for calculating te polar coordinates of te position vector ρ as a function of time since perielion. Te amplitude of te position vector is given by and its angle by ( e ) a ρ = () + ecosϑ ϕρ ϑ. () were a is te semi-major axis and e and ϑ are respectively te eccentricity of te ellipse and te true anomaly. Our metod ere, after obtaining in tis way te position vector ρ, wic is a periodic function of time, starts wit a Fourier analysis of te motion, according to te Fourier teorem. Tus, te position vector is expressed as a Fourier series given by (all te vector quantities are represented on a complex plane α β ) iωt ρ = ρ e = ρ (3) = = For circular and elliptical trajectories, te real and imaginary parts of ρ are sinusoidal and periodical non-sinusoidal oscillations, respectively. ow, Equation (3) is expressed in terms of zero, positive and negative sequence components, canging te summation limits [6]: = = ρ = ρ + ρ + ρ = ρ + ρ + ρ (4) + Tus, te following identities sow te position vector in terms of Kepler s equation (left-and side) and Fourier series (rigt-and side): ( e ) a ρ = mag ( ρ), (5) + ecosϑ ρ, (6) ϕρ ang ( ) DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

4 were i( ϕρ) + i ( ωt+ ϕρ0) + i ( ωt) ρ = ρe = ρe = ρe, i( ϕρ) i ( ωtϕρ0) i ( ωt) ρ = ρ e = ρ e = ρ e and ω is te fundamental frequency in rad/s. Te vectors ρ + and ρ are rotating vectors wit constant magnitudes and constant angular velocities, i.e., tey describe uniform circular trajectories, te first one rotates on te counterclockwise direction, wit angular velocity ω, and te latter on te clockwise direction wit angular velocity ω. Te Fourier coefficients ρ + and ρ are respectively te positive and negative armonic pasors. Te sum of tese two vectors ρ = ρ + ρ (7) + is anoter rotating vector, but wit varying magnitude and angular velocity, wic describes an ellipse illustrated in Figure (Appendix A). Tus, Equation (4) can also be written as + ( ) ρ = ρ + ρ + ρ = ρ + ρ, (8) 0 0 = = wic sows tat an elliptical trajectory described by te position vector ρ is a composition of elliptical armonic trajectories. In turn, te velocity vector can be obtained in terms of ρ + and ρ (Appendix B) as were k = ω. + d d d V ρ ρ ρ = = + dt dt dt + dρ = V + V = + dt d ρ dt + = = + + iωρ iωρ ik ρ ik ρ = = = = = =, (9) Figure. Illustration of te position vector trajectory for armonic order, sowing te positive and negative sequence motions, for te case of zero value for teir initial angles. DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

5 Te velocity vector is usually obtained as te time derivative of te position vector from Clarke s equations, but its magnitude can be expressed by te Vis-viva equation; now it can also be obtained as a summation of terms wit te armonic position vectors ρ + and ρ. Tus, its magnitude and angle can be given by te following identities: V = GM mag ( V ), (0) ρ a V ( ) ϕ ang V. () π ote tat ϕv = ϕ ρ +, and tat tis ortogonality is related to te armonic reactive power concept presented in capter III. Te total velocity V is not ortogonal to te total position vector. Suc properties from elementary kinematics are studied in detail in te same capter, concerning te powers involved, and new power concepts are ten presented. And te gravitational force, in turn, is obtained from ewton s universal law equation, and now, alternatively, in terms of te armonic position vectors ρ + and ρ dv + dv dv F = m = m + dt dt dt + = m ωρ + ωρ () = = + = m k ρ + k ρ. = = Tis conclusion stablises tat, on orbital motions, ewton s universal gravitational law, also known as ewton s inverse-square law, and Hook s elastic law can now be unified and written as an identity: GMm i F ϕ G e ρ + = m k ρ kρ ρ + = =. (3) Hook s law is evidenced on te rigt-and sides of tese identities, because eac armonic force component is proportional to te magnitude of te same armonic order position vector, were Hook s elastic constant is k = mk. Tus, gravitational forces can be given as a weigted sum of armonic positions vectors. Tis identity can also be interpreted as a statement tat te gravitational force is a summation of armonic centripetal forces. To obtain te armonic pasors ρ +, te ISV ρ is rewriting on new complex d q coordinate systems determined by te ortogonal unity vectors e i ω u t d = and u q π e i ω t + =, wic rotates on counterclockwise direction wit angular velocity ωω, were ω t is measured from te real axis α from te α β stationary coordinate system and, for ρ, d-q rotates on clockwise direction. DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

6 On tese new coordinate systems, eac armonic rotating vectors of order, ρ + and ρ, become te stationary vectors ρ + and ρ, usually called pasors, and all te oter rotating vectors of different orders ave zero mean values of its real and imaginary parts. Te fundamental angular velocity ω (fundamental frequency) is obtained from te tird Kepplers law, wic gives te orbit period D. Tis metod is general for any closed elliptical orbit, considering tat for eac orbit, wic are determined by te amplitude of its semi-major axis, its eccentricity, te mass of te central and te orbiting bodies, te gravitational constant and te period, te fundamental frequency in Hz is calculated as mentioned above, and te time step is cosen suc as to avoid aliasing. 3. Energy and Power Analysis An ideal orbital motion is an autonomous conservative (non-dissipative) second order system, i.e., a system wit no external excange of energy, wit two internal storage energy elements: te gravitational field and te moving mass. Considering tis motion as a composition of two ortogonal motions, te kinetic and potential energies are excanged between tese two motions, in tis way (Appendices C and D): For ideal circular motions, tere is an excange of kinetic energy between te two ortogonal motions, as well as of potential energy, but not between tese two forms of energy, suc tat te sum of te kinetic energy and te sum of te potential energy are constant. Tat oscillating energies are related to force and velocity in quadrature and teir rate of oscillation is called ere imaginary power. For ideal elliptical motions, in turn, tere is also an excange of kinetic energy between te two motions, as well as of potential energy, and an excange of kinetic energy from one motion wit tat potential energy from te oter motion, suc tat te sum of te kinetic energy and te sum of te potential energy are not constant. Tat oscillation of energy between te two forms is related to force and velocity in te same direction (but not in pase) and its rate of oscillation is called ere real power. Te amount of tis latter oscillating energy depends on te trajectory eccentricity. Tere is no a consensual interpretation for te analogous forms of tose energies on electrical circuits analyses [7]. 3.. Kinetic Energy Considering te armonic elliptical motion of order, te correspondent armonic kinetic energy, in terms of sequence components, is K = mv = mv V = m ( V ) + ( V ) + V V cos( ϕv ϕ V ) (4) Te average value of tis energy can be given as DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

7 were K + Kavg = K = mv (5) + + K = K + K, (6) + Q = mv =, (7) + + ( ) ω K Q = mv =, (8) ( ) ω and Q +, Q are respectively te positive and negative sequence armonic imaginary (reactive) powers. ow, in terms of sequence components, te total kinetic energy of te motion is given as were + + i V V e ϕ V + = + + = = = ( + )( + ) = m ( V ) + ( V ) + VV + VV (9) + + = m ( V ) + ( V ) + VV cos ( ϕv+ ϕv), K mv mvv m V V V V i and V V e ϕv =, werein bot magnitudes and bot angles vary at variable rate. as Te average kinetic energy of te total motion during one cycle, can be given ( + ) K = K + K. (0) av = Te rate of te total kinetic energy variation is te total power excanged between te gravitational field and te moving mass. So far, we ave seen tat te elliptical motion of a satellite around a central body is composed of a summation of armonic elliptical motions, eac one concerned to te respective armonic position vector. For every armonic motion, tere are potential and kinetic energies. Part of tese energies is constant and anoter part oscillates between tese two forms of energy. Te amount of oscillating energies depends on te trajectory eccentricity. For example, a circular trajectory (e = 0) as only constant energy stored, i.e., tere is no energy excanged between te potential field and te moving mass. 3.. Potential Energy Considering te gravitational potential energy referenced to a point at infinity, U = F ρ () G G and te potential elastic energy, referenced to te origin of te coordinate system, UHook = FG ρ, () DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

8 te gravitational energy, wic is te sum of te kinetic and potential gravitational energies, E = K + U, (3) G were E G < 0 and U G < 0 ; and te elastic energy, wic is te sum of te kinetic and potential elastic energies, E = K + U, (4) Hook were E Hook > 0 and U Hook > 0, are related in te following way: EG = EHook, (5) were E Hook = G Hook Hook = E (6) and And, from (3), we ave EHook = K + UHook, (7) F E = F ρ. (8) Hook = mk ρ (9) 3.3. Power To deal wit gravitational energies and teir rate of variation in terms of Hook s equations, it is adopted in tis work te instantaneous complex power (ICP) approac [3] [4] [5] [6] [8] [9]. Tis approac elps understanding ow eac armonic motion contributes wit te total power (see Appendix D). For eac armonic pair of force and velocity, te ICP is + + ( )( ) ( ) ( ) ( ) ( ) S = FV = F + F V + V = F V + F V + F V + F V = S + S + S + S. Te real part of S + and S, P + and P, respectively, are te positive and te negative sequence real powers and are nil in te case of non-dissipative motions (for example: no atmospere resistance), and te imaginary part, Q + and Q, respectively, are te positive and te negative sequence imaginary powers and tey are constant. Te remaining terms of Equation (30) are related to te power oscillation between te kinetic form and te gravitational form for eac armonic order motion. Tus, te subtotal power oscillation due to all armonic pairs is (Appendix D) ( ) ( ) (30) Q P = Q P + Q P = Real F V + F V. (3) DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

9 Tere are also oscillating powers between te potential field and te moving mass due to forces and velocities of different armonic orders (Appendix D). Q = Q + Q ± Pij Pij Pij i= j= j i ( ) ( ) ( ) ( ) = Real Fi Vj Fi Vj Fi Vj Fi V (3) j Te total power oscillation between te gravitational potential field and te moving mass, wic is te time derivative of te total kinetic energy is given by 4. Case Study Q = Q + Q = Q + Q + Q + Q. (33) ± ± P P Pij P P Pij Pij In tis capter, tere are sown te discrete complex Fourier (DCF) analysis and syntesis results concerned to one cycle of variation of te position, velocity and force vectors. All te numerical operations were algebraic, developed in time domain. Initially, te original position, velocity and force vectors are computed using te conventional laws for orbital motions, i.e., Kepler s equations, te Vis-viva equation and ewton s law for universal gravitation, respectively, except te velocity vector angle, not obtained from Vis-viva equation. Te used approximation for obtaining te position vector is tat presented in [0]. Ten, for validating te proposed teory in tis work, a Fourier analysis is performed on te original position vector, ten it is syntetized, as well as, te velocity and force vectors, using te rigt-and sides of te respective identities (5), (6), (0), () and (), for one cycle of te orbit, and compared to tose on te left and sides. All te results were computed using Google s Drive spreadseets. Tose obtained from te conventional metods will be referred to as Kepler s results, or original results. Tose obtained by Fourier analysis and syntesis will be referred to as Hook s results, or computed results. Te orbital motion studied is a satellite s low eart orbit motion, wit te following parameters: mass m = kg, te semi-major axis a = m, te initial instant t 0 0 = 0s, te initial eccentric anomaly E 0 = 0, te gravitational 3 constant multiplied by te eart mass GM = m s te calculated period is P = 7.08 s, and te fundamental frequency is ω = rd a s. 4 Te calculated total energy (gravitational energy) using Vis-viva equation is E = J ; te calculated average kinetic energy using equation (0) is G av K = kj (error % = 0.00) and te position vector is sampled at constant rate wit time step s, wit a total of 84 samples/cycle. Te following results, computed for e = 0.3 (only for analytical purpose, because for tis eccentricity te satellite falls on te eart), sow ow te position vector at Perielion and Apelion is composed by te terms of equation (4). Referring to tis equation, it was verified tat all te armonic position vectors line up at Perielion and every even armonic position vector subtracts at te Ape- DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

10 lion, sifting, in tis way, te center of te ellipse from te center of te coordinate system. Te total sifting is due to tis one and to te sifting due to te zero-sequence vector in opposite direction, i.e., te Apelion direction. Starting wit te focus distance c = km from c = ae, te position vector magnitude at Perielion is ρ ϕ ρ = 0 = a c= km, and position vector magnitude at Apelion is ρ ϕ ρ = 80 = a+ c = km. Te computed Hook s position vector magnitude at Perielion ρ ϕ ρ = 0 = km (error % = 0.06); and te computed Hook s focus distance c= a ρ ϕ ρ = 0 = = km (error % = 0.4). Te position vector magnitude at Perielion of te computed ellipse, excluding te zero-sequence position vector ρ 0 is ( ρ ρ 0 ) ϕ 0 ( ρ ) = = = = km ρ ϕρ = 0 were te zero-sequence position vector was computed as i( ϕ 0 ) i( 80) ρ0 = ρ 0e ρ = e km. And te computed position vector magnitude at Apelion, in te same way, is ( ρ ρ0 ) ϕ 80 ( ρ = = ) = = km ρ ϕρ = 80 (error % = 0.0). Tus, by oter way, te position vector magnitude at Perielion of te computed ellipse including te zero sequence vector is ( ) ρ = ρ + ρ = = km ϕρ = 0 = 0 ϕρ = 0 So, as we ave mentioned above, tese results sowed ow te position vector at Perielion and Apelion is composed by te terms of Equation (4). Table sows te computed complex Fourier sequences components (Fourier coefficients) of Hook s position vector for tree eccentricity values (illustrated on Figure 3 and Figure 4). ote tat te angles for eac sequence component armonic are equals, and all te angles are multiples of tat one of fundamental frequency. In oter words: Te initial armonic pase angles for te displacement vectors are multiple of te mean anomaly M: ϕρ0 = ϕρ0 = M + Table sows te results for te angular momentum calculation, for tree orbit positions and e = 0.3. Tese results sow te precision of te metod on te computation of te position and velocity vectors, wic can also be seen in te figures aead. Table 3 was obtained for tree orbit positions and tree eccentricity values. Te following computed figures are referred to as original for Keplers results and computed for Hook s results. Tey illustrate te principal caracteristics of te studied orbital motion and sow te numerical precision of te results, for eccentricity e = 0.3. Figure sows te satellite s trajectory, i.e. te position vector variation; Figure 3 and Figure 4 sow te armonic vector spectrum (obtained from te results of Table ). DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

11 Table. Computed complex fourier sequences components (fourier coefficients) of Hook s position vector for tree eccentricity values. Position vector Amplitude (m), Angle (degrees) Positive sequence egative sequence e = 0., Mo = 9.0 e = 0., Mo = 8.0 e = 0.3, Mo = 7.0 e = 0., Mo = 9.0 e = 0., Mo = 8.0 e = 0.3, Mo = E+06/ E+06/ E+06 /7.0.00E+04/ E+04/ E+04/ E+05/ E+05/6.0.E+06/ E+0/8.0.70E+03/ E+03/ E+04/7.0.5E+05/ E+05/.05.88E+0/ E+0/ E+03/ E+03/ E+04/ E+04/ E0/ E+0/ E+0/ E+0/ E+03/ E+04/ E0/ E+00/ E+0/ E+0/ E+0/ E+03/ E0/ E+0/ E+03/ E+00/ E+0/ E+03/49..6E0/ E+00/ E+0/ E0/ E+0/ E+0/56. 7.E04/ E0/ E+0/ E0/ E+00/ E+0/63.4.9E04/8.05.3E0/ E+00/ E03/ E+00/ E+0/ E05/ E0/ E+00/ E04/ E0/88..58E+0/77.7.8E06/ E03/ E0/ E05/ E0/ E+00/84.8.8E07/ E03/ E0/ E0/04.3.4E+00/ E05/04.4.7E0/ E03/.4 4.5E0/ E06/ E03/ E05/0.4.00E0/ E06/0.0.49E03/05.3 Table. Computed Hook s angular momentum (e = 0.3) True anomaly (rad) Angular Momentum (mm kkkk/ss) ,889,67, ,88,03, ,877,893,36.53 Table 3. Computed Hook s results for te position, velocity e force vector for tree positions of te position vector and tree eccentricity values. Position vector-amplitude (m), Angle (degrees) e = 0. Mo = 9.0 e = 0. Mo = 8.0 e = 0.3 Mo = 7.0 Velocity vector-amplitude (m/s), Angle (degrees) e = 0. Mo = 9.0 e = 0. Mo = 8.0 e = 0.3 Mo = 7.0 Force vector-amplitude (), Angle (degrees) e = 0. Mo = 9.0 e = 0. Mo = 8.0 e = 0.3 Mo = E+06/ E+06/ E+06/ E+03/ E+03/ E+06/ / /9.3.64/ E+06/ E+06/ E+06/ E+03/ E+03/ E+03/ / / / E+06/.5 6.5E+06/ E+06/ E+03/ E+03/ E+03/ / / /09.9 Figures 5-7 sow, respectively, te magnitude variation of te position, te velocity and te force vectors, during one cycle, for te computed values from Kepler s equations (original), compared wit tose obtained from Hook s (computed). DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

12 Beta axis (m) Alpa axis (m) Figure. Keppler s trajectory. 9.00E E E E+06 m 5.00E E E+06.00E+06.00E E Harmonic order Figure 3. Position vector spectrum. m.00e E E E E E E E+04.00E+04.00E E Harmonic order Figure 4. Position vector spectrum (negative sequence). Figure 8 and Figure 9 sow, respectively, te energy balance for te gravitational and elastic systems for instantaneous quantities computed at te initial DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

13 .0E+04.00E E+03 km 6.00E E+03.00E E True anomaly (rad) Original Computed Figure 5. Position vector magnitude variation..0e+04.00e E+03 m/s 6.00E E+03.00E E True anomaly (rad) Original Computed Figure 6. Velocity vector magnitude variation. Figure 7. Force vector magnitude variation. instant tt 0, according te equations presented in Capter III, were (a) kinetic, (b) average kinetic, (c) potential, (d) average potential, (a + c) total energy, (b + d) average total. DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

14 k J K (a) K av(b) U (c) U av(d) E (a+c) E (b+d) Figure 8. Gravitational energy k J K (a) K av(b) U (c) U av(d) E (a+c) E (b+d) Figure 9. Elastic energy. Te computed kinetic energy sown in Figure 0 was obtained using Equation (9) and compared wit tat one obtained using te velocity from Vis-viva equation. Its average value was obtained using Equation (0). Figure sows te computed kinetic power obtained using Equation (33) and compared wit te time derivative of te original one, obtained using te Vis-viva equation. Figure sows te gravitational potential energy variation (Equation ()) using Keppler s equations and Hook s. 5. A Proposition for te Starting Point for a Practical Syntesis of te Quantities Related to Orbital Motions As we ave seen tat closed gravitational trajectories on a plane can be interpreted as a composition of periodic motions on eac axis of a system of ortogonal coordinates, generating te known Lissajous figures, we can extend tis concept for motions on tree symmetrically displaced axes, named pase, a b and c axes, given respectively by te unity vectors a 0, a and a were DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

15 k J 50, , , , , , , , , , True anomaly (rad) Original Computed Average Figure 0. Kinetic energy variation. W 5, , , , , , , , , , True anomaly (rad) Original Computed Figure. Kinetic power variation. 0.00E E E E+07 J -4.00E E E E E+07 True anomaly (rad/s) Original Computed Figure. Gravitational potential energy variation. π 3 a = e i, on a plane (more axes are possible too), using te inverse transforma- tion of Clarke s transformation DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

16 ρ = ( ρa + aρb + a ρc) = ρα + iρ, (34) β 3 were ρ a, ρ b and ρ c are periodical oscillations respectively on eac axis, called pase quantities. To do tat inverse operation, knowing te position vector ρ, its projection on axes a b and c gives, respectively, te scalar quantities ρ a, ρ b and ρ c [6]. Looking te position vector in tis way, tey can be called instantaneous space vectors (ISVs), by analogy wit electrical polypase circuits, were tey can represent voltages, currents and fluxes (te correspondent electrical variables are respectively carge for position in te mecanical system ρ q, current for velocity v i and voltage for force f v). Te property of ISVs carrying positive, negative and zero sequence components were studied by Akagy, Ferreiro, Willems, Emanuel and te autor [3] [4] [5] [6] [8] [9]. But, in fact, tat property, for only te cases of positive and negative sequences, is evidenced in te terns of te Fourier series. Fortescue deals wit tose components on pase analyses of unbalanced tree-pase electrical circuits, Clarke wit bot te tree-pase and te equivalent two-pase circuit obtained after Clarke transformation, and te complex Fourier series approac deals only wit te equivalent two-pase circuit, witout mentioning te possibly relations wit n-pase oscillating systems. Tus, aving stablised tose relations between tree-pase systems and te equivalent two-pase systems, syntesis of ISVs may be carried out using power electronics tecniques and electromecanical devices. In tis way, many alternatives can be proposed for future researc; more precisely, acting wit conventional propulsion systems or ybrid electromecanical systems aving in mind tat eac armonic variable of positive and/or negative sequence can be controlled individually or in armonic ranges. Using equivalent tree-pase mecanical or electrical systems (motions in tree symmetrical directions), te kinetic and potential stored energies are respectively given as ([6], Appendix) 3 3 K = mv EL = LI, (35) U Hook = FG ρ EU = CV, (36) 4 4 were E L, L and I are respectively magnetic energy, inductance and current and E U, C and V, respectively, potential energy, capacitance and voltage. Tose energy equations sow tat te oscillating variables on eac direction (pase on electrical systems) of a tree-axis mecanical motion ave teir magnitudes decreased, wic means material savings and more energy storage capacity. More precisely, speaking of a spacecraft, eac armonic motion following an elliptical trajectory, composed of positive and negative sequence motions, may be canceled by an opposing motion produced, for example, by te composition DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

17 of te motions of two internal rotating bodies (eac rotation in te conventional manner to produce artificial gravity and attaced to te center of te spacecraft by means of an elastic spring or an equivalent solenoid electromagnetic damper), wic rotates in opposite directions to cancel, respectively, te corresponding positive and negative armonic motions of te spacecraft or, for a more practical solution, eac armonic trajectory wit its correspondent variables may be obtained using two ortogonal coordinated internal armonic motions, or tree symmetrical coordinated internal armonic motions, produced for example by two (or tree) magnetic pistons moving inside a cylindrical electrically controlled solenoid (a kind of active linear solenoidal dampers) or linear electromagnetic motors. For tese cases, Equations (35) and (36) may be considered. Tis study is in te field of mecanical vibrations and te resonance penomena applied to te motions of two bodies, joined by an elastic device, moving in a gravitational field. All te above mentioned process is a tuning process between te cosen natural frequency motion of te spacecraft and te artificial vibrating system. Te complex geometry of te total motion as well as of eac armonic component must be carefully considered. Tis metod must be applied first to armonic motions of ig frequency, since te amount of energy is smaller tan tose of low frequencies, as well as teir amplitudes and centripetal forces; wic implies easier practical implementation, to cancel te gravitational components of a range of armonic motions. In tis way, te spacecraft s original elliptical motion takes anoter elliptical motion wit less armonic contents. Te greater te eccentricity of te orbit, te greater te amount of armonic energy tat can be dealt wit using practical applications, because, as we ave seen, armonic frequencies increase, wose correspondent trajectories are smaller; wat decreases te dimensions of te electromecanical systems. Additionally, te results sown in tis work point to te possibility tat a free fall of an object close to te eart surface, because, due te tangential velocity of te eart location, tey follow an elliptical trajectory wit eccentricity approacing to one (before itting te ground). In tis case tere must be armonics of ig orders, wic may be tuned wit electromecanical devices. All tese considerations are all ypotetical and must be proven practically. For applications near te eart surface tere are ancorages on conventional ancor foundations. 6. Conclusions A numerical simulation of an elliptical orbital motion was implemented in tis work, using Fourier analysis and syntesis of te related mecanical quantities, as well as, an energy and power study. Te great differential of te used metodology was te complementary use of Fortescue and Clark s metods of analysis. DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

18 Te results confirmed a new concept for gravitational forces, wic stablises tat ewton s Universal Law for gravitational forces on orbital motions can be considered as a sum of armonic forces of Hook s elastic type, wic is a weigted series of armonic position vectors. Te two summations on te rigt-and side of Equation (4) sow tat te elliptical motion of a satellite around a central body is composed of armonic elliptical motions, eac one produced by te composition of positive sequence and te negative sequence circular trajectories. Te metod ere employed is also an alternative way to calculate te orbital velocity magnitude and angle. Te numerical results sown in te tables, as well as in te figures, validated te metod wit good precision troug te comparison of all te quantities produced from Kepler and ewton s law and tose produced in terms of armonic vectors despite of limitations of te computational means employed. In tis way, it was numerically verified tat ewtons inverse square law for gravitational forces, on orbital motions, can be tougt as a series of armonic forces of Hooks elastic types or, in oter words, as a weigted series of armonic position vectors. Te following properties were verified for tis type of motion: Te number of armonics depends on te ellipses eccentricity. Te initial armonic pase angles for te displacement vectors are multiple of te mean anomaly M: ϕ + ρ0 = ϕρ0 = M Te Vis-viva equation as well as ewton s inverse square law equation can be, alternatively, replaced by Equations (0) and (3), respectively, for obtaining te correspondent quantities. Te Vis-viva equation gives only te magnitude of te velocity vector, but Equations (0) and () give bot its magnitude and its angle. It was observed, wit te adopted approximation [0], tat te negative sequence vectors magnitudes, as well as te armonic components of te Fourier series, increase wit te ellipses eccentricity for simulations wit eccentricity varying between zero and 0.3. Tus, it is expected tat tis tendency is true for eccentricity values tending to (radial trajectory), wic is almost a rigt line trajectory and te number of armonics tends to infinity, and te positive and negative sequence vector tend to equality. Tis case can be tougt as a free fall of a body. If tose ypoteses are confirmed, tis metodology can be extended to non-periodical motions, using Fourier transform and te armonic spectrum is almost symmetrical regarding to te axis passing troug zero frequency. Also, tis metodology can be extended to te cinematics and dynamics of any planar periodical, and, if te above ypotesis is confirmed, non-periodical motions. Moreover, an analogy between tree-pase electrical circuits and gravitational motions was evidenced. Indeed, te teory ere presented is implicit on te DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

19 study of power and energy stored on capacitor banks. Tus, te same considerations can be done for electrical systems, concerning tose quantities: te stored energy on a capacitor bank can be given as te product of voltage and electrical carge, sifting in tis way te reference point from te coordinate system origin to infinity. Tose conclusions bring fort te idea tat gravitational forces can be obtained by, for example, electromecanical devices and, in tis way, new tecniques for satellite navigations can be developed, as well as new forms of gravitational energy extractions. References [] Fortescue, C.L. (98) Metod of Symmetrical Coordinates Applied to te Solution of Polypase etworks. AIEE, 37, [] Clarke, E. (943) Circuit Analysis of AC Power Systems. Wiley, Hoboken, Capter 0, 308. [3] Akagi, H., Kanazawa, Y. and abae, A. (983) Generalized Teory of Instantaneous Reactive Power in Tree-Pase Circuits. IPEC-83, Tokyo, [4] Ferreiro, A. and Superti, G. (99) A ew Approac to te Definitions of Power Components in Tree-Pase Systems under onsinusoidal Conditions. IEEE Transactions on Instrumentation and Measurement, 40, ttps://doi.org/0.09/9.870 [5] Willems, J.L. (99) A ew Interpretation of Akagi-abae Power Components. IEEE Transactions on Instrumentation and Measurement, 4, ttps://doi.org/0.09/ [6] Milanez, D.L. and Emanuel, A.E. (003) Te Instantaneous-Space-Pasor a Powerful Diagnosis Tool. IEEE Transactions on Instrumentation and Measurement, 5, [7] Emanuel, A.E. (00) Power Definitions and te Pysical Mecanism of Power Flow. IEEE Press and Wiley, 64. ttps://doi.org/0.00/ [8] Gossick, B.R. (967) Hamilton s Principle and Pysical Systems. Academic Press, ew York and London, 47. [9] Akagi, H. and abae, A. (983) Te p-q Teory in Tree-Pase Systems under on-sinusoidal Conditions. European Transactions on Electrical Power, 3, 7-3. ttps://doi.org/0.00/etep [0] Randal, D.P. (99) True Anomaly Approximation for Elliptical Orbits. Journal of Guidance, Control, and Dynamics, 4, ttps://doi.org/0.54/ DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

20 Appendix A Te armonic component of position vector were expressed on Cartesian coordinates, ρ = ρ + ρ, i( ϕρ) + i ( ωt+ ϕρ0) + i ( ωt) ρ = ρ = ρ = ρ e e e, i( ϕρ) i ( ωtϕρ0) i ( ωt) ρ = ρ e = ρ e = ρ e, ( t 0) isin ( t 0) ρ ρ ( t ρ 0) isin ( t ρ 0) ρ = ρ cos ω + ϕ + ω + ϕ + ρ cos ω ϕ ω ϕ + at te perielion position were ϕ 0 = ϕ 0 = 0, results tat ρ ρ + + ( ) cos( t) i( ) sin ( t) ρ = ρ + ρ ω + ρ ρ ω. + ow, considering tat ρ + ρ = a, te semi-major ellipses axis, and ρ =, te semi-minor axis, te ellipses condition, + ρ b is fulfilled. Appendix B ( ω ) sin ( ω ) a cos t b t + = a b Te following are te expressions of te positive and negative sequence armonic position vectors: i( ϕρ) + i ( ωt+ ϕρ0) + i ( ωt) ρ = ρ = ρ = ρ e e e, i( ϕρ) i ( ωtϕρ0) i ( ωt) ρ = ρ e = ρ e = ρ e, teir first derivatives + d ρ + i ( ωt) + = iωρ e = iωρ, dt d ρ = iωρ = dt i ( ωt) e iωρ and second derivatives + d ρ d dt = + iωρ, dt d ρ d dt = iωρ. dt DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

21 Appendix C Te constant gravitational energy is te sum of te instantaneous kinetic and potential energies, and equal to te sum of teir average values: EG = K + UG = K + UG. In turn, te average value of te kinetic energy is alf te average potential energy value, wit opposite sign: K = U. Tus, For te constant elastic energy, G K = E G. EHook = k + UHook = K + UHook. But, in tis case, te average value of te kinetic energy is equal to te average potential energy value, Terefore, K = U Hook. K = E Hook As te kinetic energy as te same value on any reference system, comparing te kinetic energy of bot systems, imply tat EG = EHook. Q.E.D. Appendix D In tis Appendix, we are going to see ow is te power contribution of every armonic motion (positive and negative sequences armonic motions) to te total power given as te derivative of kinetic energy given by Equation (4); separating tem as armonic powers related to forces and velocities of same armonic order, and armonic powers related to forces and velocities of different armonic orders. Te instantaneous complex power (ICP) concept is presented in [4] [5] [6] [8] [9]. ICP is a time-varying complex quantity given by ere + + ( )( ) S = FV = F + F V + V = FV + FV + FV + FV dv + F + = m and d t dv F = m d t DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

22 D. Harmonic Powers Due to Forces and Velocities of Same Harmonic Order For eac armonic pair of force and velocity, te ICP is + + ( )( ) ( ) ( ) ( ) ( ) S = FV = F + F V + V = F V + F V + F V + F V = S + S + S + S + ± Te first two terms on te rigt-and side of tis expression, S + and S, are constant and are called complex powers, and te oter two, S ± and S are oscillating powers, and are called ICPs. Te real part of S + and S, P + and P, respectively, are te positive and te negative sequence real powers and are nil in te case of non-dissipative motions (for example: no atmospere resistance): P + + P ( ) 0 ( ) 0 = Real S = = Real S = and te imaginary part, Q + and Q, respectively, are te positive and te negative sequence imaginary powers and tey are constant: ( ) ω ( ) Q = Imag S = m V ( ) ω ( ) Q = Imag S = m V Real powers are related to force vectors in pase wit velocity vectors, and imaginary powers are related to tose vectors in quadrature. As we are interpreting te orbital motion as two coordinated motions, on two ortogonal axes on a plane, te pysical meaning of tese imaginary powers is tat tey are te pick values of te rate of kinetic energy excanged between te two axes, wic are sinusoidal quantities, respectively on te equivalent positive and negative sequence motions (te same interpretation is used to define reactive powers on electrical circuit analysis). ote tat tey are related to te armonic sequence kinetic energies in tis way: K K ( ) + Q = mv = ω + + ( ) Q = mv = ω were K + and K are, respectively, te kinetic energy associated to te armonic positive and negative sequence motions. Tey are constant quantities. Te sum of tese two armonic kinetic energies given by K ± = K + K + is te average kinetic energy during a cycle, related to tis armonic motion (see DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

23 equation 5): Te amount ± Kavg = K = mv ( + ) K = K + K av = is te average kinetic energy of te total motion during one cycle. Considering now te ICP terms S ± and S of te expression for S, we ave tat, as in te cases of te above mentioned real and imaginary powers, teir real Q ± P, Q P and imaginary Q ± Q, Q Q parts are also results of force and velocity components, respectively in pase and in quadrature. But, ere, tese components are not constant during a cycle of te motion, i.e., tey vary wit frequency ω, and teir average values are nil (in an analogous electrical circuit, due te fact tat te elements of te analyzed system are storage energy, or reactive, elements, even been powers produced by voltages and in-pase current vectors, tese powers are named instantaneous reactive powers, or nonactive powers ([7], p. 56). Te term inactive is due to te fact tat te power flow is a system internal power flow. Tus, te same denominations are adopted ere. It can be verified for tese armonic instantaneous reactive powers tat Q ± = Q and P P Q ± + Q = 0. Q Te pysical meaning of tis is tat tere is a power oscillation between te kinetic form and te gravitational form for eac armonic order motion, given by Q Q ± ± = Q + Q = Q. P P P P Tus, te subtotal power oscillation, wic is due to all armonic pairs is ( ) ( ) Q P = Q P + Q P = Real F V + F V. D. Harmonic Powers Due to Forces and Velocities of Different Harmonic Orders Tere are also oscillating powers between te potential field and te moving mass due to forces and velocities of different armonic orders. Let us represent tem by and, given respectively by S ± ij S ij ( ) ( ) S F V F V ± ij = i I + i I and ( ) ( ) S F V F V + ij = i I + i I DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology

24 Te real part of tese powers Q = Real S and ( ij ) ± ± Pij Qij ( ij ) Q = Real S are te rate of energy excanged between te moving mass of te armonic motion of order i and te potential field of armonic motion of order j. Tus, te subtotal power oscillation due to all armonic of different orders is Q = Q + Q ± Pij Pij Pij i= j= j i ( ) ( ) ( ) ( ) = Real F V + F V + F V + F V i j i j i j i j and te total power oscillation is te summation of tese two subtotal powers Q = Q + Q = Q + Q + Q + Q ± ± P P Pij P P Pij Pij OTE : te zero-sequence vector as anoter meaning in electrical circuit analysis. DOI: 0.436/jepgc Journal of Hig Energy Pysics, Gravitation and Cosmology