A NEW WAVE EQUATION FOR LAGRANGIAN DYNAMICS: THE PLANAR ORBIT FOR ANY FORCE LAW

Transcription

1 Jounal of Foundations of Physics and Chemisty () A NEW WAVE EQUATION FOR AGRANGIAN DYNAMICS: THE PANAR ORBIT FOR ANY FORCE AW M. W. Evans, H. Eckat and B. Foltz Alpha Institute fo Advanced Studies A new wave equation is developed by agangian methods fo applications in celestial mechanics as pat of ECE theoy. A method is infeed fo the calculation of a plana obit fo any foce between a mass m obiting a mass M in a plane. Two methods of solution of the wave equation ae given and the self consistency of the method checked in the Newtonian limit. It is shown that the Einsteinian geneal elativity (EGR) is one out of an infinite numbe of foce laws that give obital pecession. Keywods: ECE theoy, celestial mechanics, agangian wave equation, plana obit fo any foce law.. INTRODUCTION As pat of this seies of two hunded and foty two papes to date {-}, Einsteinian geneal elativity (EGR) has been efuted in compehensive detail and in many ways. In ode to seek a eplacement theoy applicable fo all obsevable obits, a numbe of methods have been poposed in ecent papes of this seies. It has been found that obital pecession is given by an infinite numbe of foce laws, not just the foce law of Einsteinian geneal elativity (EGR). Fo example the Minkowski foce gives a pecession, and this pocess has been animated by B. Foltz on Any foce law popotional to to the powe n gives a pecession, with the sole exception of n =, and this pecession has again been animated on by B. Foltz. Essay 8 on lists fifty seven of the available efutations

2 M. W. Evans, H. Eckat and B. Foltz of EGR, so it is a completely efuted theoy, and the peceding papes of this seies seveely citicise the dogmatic claims of EGR to be a pecise theoy. In Section a wave equation is infeed fom a combination of the two classical Eule agange equations of an object of mass m obiting an object of mass M in a plane. These classical equations ae used in EGR, despite its claim to be a elativistic theoy. So EGR is not only incoect in many ways, it is conceptually self inconsistent. This wave equation can be extended to the Minkowski spacetime staightfowadly, and that will be the subject of futue wok, but in this pape the classical limit is examined. Two solutions ae given of the wave equation, one deived by compute algeba by H. Ecka. This second solution is tested fo self consistency and coectness in the Newtonian limit and applied with some simple foce laws to find the obit. As fa as the authos ae awae, this is the fist time that a method has been devised in celestial mechanics to find the obit fo any foce law. The obit fo the Hooke Newton invese squae law is an ellipse, as is well known. The pecessing ellipse is given by an infinite numbe of othe types of foce law. Using this method the claims of EGR can be tested in anothe way. Section is a desciption of the numeical methods used by H. Ecka: compute algeba used to check the hand calculations and used to solve the new wave equation.. WAVE EQUATION AND PANAR ORBITA SOUTIONS FOR ANY FORCE Conside the obit of a mass m aound a mass M in a plane. The two equations of motion fom a well known agangian analysis {} ae: and = m θ F () = m θ () Hee F is the foce between m and M and is the conseved total angula momentum. The plane pola coodinates ae and θ. Fom Eq. () in Eq. (): m = F () 4 m so: F = (4) m m and F ω = (5) m This is a wave equation with stuctue: d + Ω = (6)

3 whee A New Wave Equation fo agangian Dynamics = F ( ) Ω m 4 Unde a well defined constaint to be discussed below, this equation is a hamonic oscillato with peiod: m (7) T = π Ω (8) and apsidal angle: whee the angula velocity is defined by Eq. (): Theefoe the angula coodinate is: and can be found fo any foce. Eq. (6) has the solution: with: povided that: whee: θ ψ = T d = ωt (9) dθ ω = = m () θ= ψ () = exp ( i Ω t) () F T = π 4 m m dx Ω = + x= i + t d Ω Ω / In ode to pove this, note that Ω is a function of time, so: and d e d e () x (4) (5) xe Ω = (6) iωt i t dx Ω = + x e (7) i Ω t i t

4 4 M. W. Evans, H. Eckat and B. Foltz i.e. so: d dx x + = (8) Ω dx = + x (9) Q.E.D. Eq. (9) is a second ode diffeential equation fo the time dependence of Ω and is a subsidiay o constaint equation that must be obeyed if the solution () be tue. Eq. (9) can always be solved, analytically o numeically, so Eq. () is always tue. The second method of solution of Eq. (6) is due to compute algeba by H. Ecka, giving the solution: / t = ( Ω d A) d+ B () whee A and B ae constants of integation which have to be found by futhe analysis. Fo any cuve {}: da = dθ () whee A is an aea. Assume that in time T an aea A is swept out. Then in time t an aea At/T is swept out. Theefoe: and in geneal: Fom Eqs. () and (): fo any cuve and any foce law. Define the function: At / T = da = dθ () T t = A d θ () + = / T Ω d A d B A = / Ω f : d A and fo simplicity assume that B is zeo. Then: and: t = f ( ) d = d = f ( ), dθ = d θ (4) (5) T A d θ (6) T A (7)

5 Using the chain ule: and the angula pola coodinate is: A New Wave Equation fo agangian Dynamics 5 dθ dθ = = d d θ= A T f f A T (8) d (9) and can be found fo any foce law, Q. E. D. Fo any cuve in the classical agangian dynamics {} of any plana obit: Integating: so: T m = da A = T m fo all plana obits. In the Newtonian limit {-} the foce law is: () m A = da () F ( )= αm whee α is the half ight magnitude of the elliptical obit: = + α cosθ whee is the eccenticity. It was checked by compute algeba as follows that the esult (9) gives the ellipse (4) fo the Newtonian foce law (). This is a igoous test of the self consistency of the theoy. The solution of Eq. (9) with the foce law () is, by compute algeba: () () (4) α ma θ = y sin (5) T ( α Am ) / whee y is a constant of integation. If: A = ( ) + α m and (6) y =π (7)

6 6 M. W. Evans, H. Eckat and B. Foltz then: i. e. Using Eq. (), Eq. (9) simplifies to: α ma θ= π sin / T (( ) ( + )+ ) (8) T = α + cos θ ma (9) α = (4) + cosθ which is Eq. (4), Q. E. D. So Eq. (9) is igoously coect fo the Newtonian foce. Having tested it in this way it may be applied to othe foces and this is done in Section. Eq. (6) can be simplified by noting that: α α a = = (4) ( + ) whee a is the semi majo axis of the ellipse {}. So: In the Newtonian theoy {}: A aαm = (4) so the constant of integation x is: = m MGα (4) which has the coect units of the squae of linea velocity. The pecessing conical section: α cos xθ is found fom the foce law {-} A = MG (44) a = + (45) mmgx mmg F= α ( x ) (46) and compute algeba is used in Section to check this esult, giving a pecise esult fo the pecession constant x, a new esult and an advance ove pevious wok. EGR claims that the foce law is:

7 A New Wave Equation fo agangian Dynamics 7 mmg MG F= 4 mc This claim can be tested by using Eq. (47) in Eq. (9) to find the pecession of the pola angle due to EGR, which claims that fo a evolution of π: whee: (47) θ π + x (48) x GM = ac This claim of EGR is based on the same agangian analysis that leads to Eq. (6), so Eq. (9) must poduce the esult (49) if EGR is coect. It is known fom Essay 8 on that EGR is incoect in many ways. Finally in this Section the Minkowski foce has been shown in immediately peceding papes to be: 4 4 γ γ F= γ m d d d e + e θ (5) m mc whee e and e θ ae the unit vectos of the plane pola coodinate system. The Minkowski foce is the Newtonian foce coected fo the oentz facto: v γ= (5) c whee the velocity appeaing in the oentz facto is given by the Minkowski metic as: Fom Eq. (): so: Using the chain ule: v d / d = + m d = f d = f ( ) d dθ = = dθ m so Eq. (55) can be used to find the obit in the fom: d d m d d dθ θ = = (49) (5) (5) (54) (55) m (56) f

8 8 M. W. Evans, H. Eckat and B. Foltz fo any foce law. Using Eq. (54) in Eq. (9) the tue anomaly (o obital pola angle) can be expessed as: whee: So the tue anomaly becomes: implying that: θ= m f d f = d A Ω / (57) (58) θ= m (59) dθ = m (6) which is Eq. () Q. E. D. So the theoy of this section is igoously coect and self consistent and is a new and poweful method of celestial mechanics.. COMPUTATION OF ORBITS FOR SEECTED FORCE AWS Befoe consideing the obits emeging fom the foce laws we calculate the apsidal angles fo some cases. The apsidal angle is defined by Eq.(9) and can be witten with aid of (7), (8) and (6): π ψ = F ( ) (6) m 4 m m The esults ae pesented in Table. In most cases the apsidal angle depends on the adius. To find a easonable value fo we calculate fo the case ψ = π The apsidal angle is constant only fo the / foce law which gives a pecessing ellipse of canonical fom. Fo highe exponents vaies as well as fo an exponential and even a constant foce law. The same holds fo a adially oscillating foce. In the following we apply Eq. (9) to some foce laws to obtain the obital function θ() in the most geneal way. With the geneal esult () this faction can be witten f ( ) θ= d + B (6) m with f = d A Ω / whee Ω is given by Eq. (7) A and B ae constants of integation. Inseting the foce law of a pecessing ellipse, (6)

9 A New Wave Equation fo agangian Dynamics 9 mmgx α x F= mmg, (64) Table Apsidal angles ψ and adii fo ψ = π fo seveal foce laws. Foce aw Apsidal angle Radius fo Ψ = π mgm F= π ψ = = αm α = α F= β ψ = π βm any F= β 4 ψ = π βm m = β β F= Fe ψ = πe m F β β e e = mf β F= F ψ = F= Fsin κ ψ = π m F π m sin ( κ) F = mf = Fmsin κ leads (by compute algeba) to the obit elation α = sin ( x ( θ + B )) + With x α Am = x x α Am = x α Am x (65) (66)

10 M. W. Evans, H. Eckat and B. Foltz By choosing the constant we can utilize the elation B = π (67) x sin β π cos β = so that we aive at the standad obit of a pecessing ellipse: = α cos ( x θ )+ The emaining constant A is elated to the physical paametes via Eq.(66): x A = α m The constaint that the squae oot in has to be eal valued, leads to the estiction o x (68) (69) (7) α Am (7) A x α m In case of equality we have a cicula obit. The limit of a paabola ( = ) is eached fo A =, and the obit is hypebolic fo A <. The obit of the pecessing ellipse is gaphed in fom θ() and dθ/d in Fig. fo compaison with othe foce laws. The vetical tangents at the etun points of ae significant. Next we inspect the Einstein foce law mmg MG F() = (7) 4 mc It follows fom Eq. (6) that the integal is not solveable analytically, it must be evaluated numeically. The obit is given by θ= c m d + B (74) c m + GM c Am cm The Minkowski foce law is ( ) 4 (7) γ mmg mm F= γ α γ( ) (75) The obit can be calculated fist in the appoximation of a constant elativistic γ facto which is defined by γ = υ (76) c

11 A New Wave Equation fo agangian Dynamics with υ being the absolute value of the obital velocity. Assuming a nea-cicula obit may justify the appoximation υ const. This esults in an obit with = sin 4 γαmm γαmm+ + 4 θ B γ γ αm M γ 4 mgm (77) = m γ 8 m GM + γ γ 4 αam M A (78) The esult is a pecessing ellipse which is defomed compaed to the canonical case of Eq.(65), see Fig. in compaison to Fig.. When the coect fom of γ is to be taken into account, we have to use an expession fo the velocity. Hee we took the appoximation fom the notes of pape 8: γ= MG c α so that we have a adial dependence only. Then the function f () can be evaluated. The stuctue of constants is so complicated that we only give the expession used fo the plot, the integand of Eq. (9): (79) m f = + log log (8) This integand epesents dθ/d and is shown in Fig. as the ed cuve. This function has been integated numeically to give the blue cuve. The gaphics can diectly be compaed to Fig. fo the constant gamma case. One sees that θ() behaves diffeently at the lowe bounday, a clea elativistic effect. It should be noted that the paametes in the fomulas wee expeimentally adopted fo the gaphs in such a way that the esults wee compaable. In addition they wee chosen so that bound cuves came out (no hypebolas etc.). It would take a geate effot e.g. to set the integation constants in a way that the same physical system is descibed in all cases.

12 M. W. Evans, H. Eckat and B. Foltz Figue θ() and dθ/d fo a pecessing ellipse with =., α =, x =.. Figue θ() and dθ/d fo the Minkowski foce with adopted paametes, γ = const.

13 A New Wave Equation fo agangian Dynamics Figue θ() and dθ/d fo the Minkowski foce with adopted paametes, numeical solution. ACKNOWEDGMENTS The Bitish Govenment is thanked fo a Civil ist Pension and the AIAS and othes fo many inteesting discussions. Dave Buleigh is thanked fo posting and Robet Cheshie and Alex Hill fo boadcasting and tanslation. REFERENCES [] M. W. Evans, Ed., Definitive Refutations of the Einsteinian Geneal Relativity special issue six of efeence, (Cambidge Intenational Science Publishing, CISP, ). [] M. W. Evans, Ed., J. Found. Phys. Chem., (CISP, Six Issues a Yea). [] M. W. Evans, S. J. Cothes, H. Ecka and K. Pendegast, Citicisms of the Einstein Field Equation (CISP, [4] M. W. Evans, H. Ecka and D. W. indstom, Geneally Covaiant Unified Field Theoy (Abamis 5 to ) in seven volumes. [5] M. W. Evans, H. Ecka and D. W. indstom, papes and plenay in the Sebian Academy of Sciences, to pesent, and papes in Physica B, Found. Phys. ett., and Acta Phys. Polonica on ECE theoy.

14 4 M. W. Evans, H. Eckat and B. Foltz [6]. Felke, The Evans Equations of Unified Field Theoy (Abamis 7, Spanish tanslation by Alex Hill on [7] M. W. Evans and. B. Cowell, Classical and Quantum Electodynamics and the B() Field (Wold Scientific ). [8] M. W. Evans and S. Kielich, Ed., Moden Non inea Optics (Wiley Intescience, New Yok, 99, 99, 997, ) in two editions and six volumes. [9] M. W. Evans and J.-P. Vigie, The Enigmatic Photon (Kluwe, Dodecht, 994 to ) in ten volumes hadback and softback. [] M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field Theoy (Wold Scientific, 994). [] J. B. Maion and S. T. Thonton, Classical Dynamics (Hacout, New Yok, Thid Edition, 988).