www.BillHowell.ca's review of Bill Lucas's math in his book the "Universal Force, Volume I" /*$ echo "version= $date_ymdhm, cos - 1 inclusion : $cos_inclusion" >>"$p_augmented" This file is : /*$ echo "$p_augmented" >>"$p_augmented" /********************** SUMMARY This file contains my own step-by step derivations of equations in Chapter 4 of Charles W. Lucas's book "The Universal Force, Volume 1". Discrepancies between Lucas's results and my own are far more likely due to my own errors than Lucas's, but there is a chance that several minor errors in Lucas's book will have been highlighted. /*$ cat >>"$p_augmented" "$d_Lucas""context/summary - general.txt" /*_endCmd /*$ cat >>"$p_augmented" "$d_Lucas""context/text editor - how to set up.txt" /*_endCmd *********************** TABLE OF CONTENTS /*_Insert_Table_of_Contents EQUATIONS : /*_Insert_equations For instructions on how to update the Table of Contents and Equations, see the section "Document build short description" at the end of this document. There is currently a problem of both lists above "shifting" the line number counts. /********************************************** waiver, copyright /*$ cat >>"$p_augmented" "$d_Lucas""context/waiver, copyright.txt" /*_endCmd /********************************************** >>> Lucas's Dedication (This is copied directly from his book.) /*$ cat >>"$p_augmented" "$d_Lucas""context/Lucas dedication.txt" /*_endCmd /********************************************** >>> Introduction /*$ cat >>"$p_augmented" "$d_Lucas""context/introduction.txt" /*_endCmd /********************************************** ; >>> Lucas 4 - Derivation of the Universal Electrodynamic Force Law for constant velocity /********************************************** ; >>>>>> 4.1 - Proper Axioms of Electrodynamics p63h0.2 "... In the derivation that follows [2,3,4] the approach is taken that a proper electrodynamic force law should be compatible with the following set of axioms which is more complete than that of Maxwell and corresponds better with reality : Lucas_ED_axioms := 1. Coulombs law for the force between static charges 2. Amperes generalized law for the force between current elements 3. Faradays law of electromagnetic induction 4. Gausss laws 5. Only contact forces exist in nature 6. Lenzs law for induction 7. Finite-size of charged particles with interior structure 8. Fields of charges remain attached when charges move and have tensile strength 9. Galilean invariance 10. Newtons third law - for every action there is an equal and opposite reaction 11. Conservation of kinetic and radiant energy 12. Conservation of momentum (radiation reaction etc) 13. Machs principle that local physical laws are determined by the large-scale structure of the universe 14. Nonlinear electrodynamic processes occur in lasers and other phenomena Note that axioms 5-14 are missing in the relativistic version of electrodynamics based on Maxwells equations. ..." /********************************************** ; >>>>>>>>> Fundamental Equations Of Electrodynamics ; p63h1.0 "... The fundamental equations of electrodynamics are based upon six empirical laws that are valid for constant velocity, i.e. Note that both Amperes law and Faradays law involve the observers reference frame and the primed moving frame of reference that are described by the Galilean transformation of Lucas04_07. 17Aug2015 Howell - variables t, c, n q are scalar, the rest are vector. Also, the primes (´) indicate the moving frame of reference, and unprimed is the observers frame. 21May2016 Howell - There are big differences between Lucas's six empirical laws, and classical conventional expressions (eg Maxwells equations). I have noted that in my comments for equations (4-1) through (4-6) below, and I will revisit Lucas's justifications in Appendix A etc later. For this first-pass-through,as with other problematic situations, I simply follow on with Lucas's [framework, basis] as a check on the [consistence, correctness] of derivations that flow from it. /********************************************************** /*------> (4-01) Generalized_Amperes_Law Equation (4-1) is DIFFERENT from Maxwell equation equivalent! - Doesn't have the curl operation on B, rate of change of E0. Also - Lucas's dividing by c, maybe μ0,ε0 are problems too, I have NOT properly adjusted formulae from various sources for differences in the units used, whether Gaussian, SI, or other. This creates some confusion here and there in my review comments.Jackson 1999 p782h0.15 Table 3 provides conversions reference Lucas p65h0.5 This is shown in (5-8), derived in Appendix A reference : https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law Maxwell-Ampere circuital equation, integral form, C is a closed curve : /$ ∮[∂(l): B) = ∬[•dArea: μ0*J + μ0*ε0*∂[∂(t): ET]] /% ∮[∂(l): B) = ∬[•∂(Area): μ0*J + μ0*ε0*∂[∂(t): E]] /* differential form : /$ ∇B = μ0*J + μ0*ε0*∂(ET) /% ∇B = μ0*J + μ0*ε0*∂(E) /*"In cgs units" ???? integral form : /$ ∮[∂(l): B) = 1/c*∬[•dArea: 4*π*J + ∂[∂(t): ET]] /% ∮[∂(l): B) = 1/c*∬[•∂(Area): 4*π*J + ∂[∂(t): E]] /*differential form : /$ ∇B = 1/c*[4*π*J + ∂[∂(t): ET]] /% ∇B = 1/c*[4*π*J + ∂[∂(t): E]] /* NOTE : It seems to be implied in the wikipedia article that [B, E] are functions of (r,t), as explicitly shown in Lucas's expressions! Lucas (4-7) below states t = t Lucas's use of the "Generalized Amperes Law" is NOT the same as the Maxwell equation equivalent. This is a major point that he is making. /*++++++++++++++++++++++++++++++++++++++ /*add_eqn 04_01 - Generalized_Amperes_Law /$ B(r´,t´) = v/c × E0(Rpcv,t´) ∇B = 1/c*[4*π*J + ∂[∂(t): ET]] (mathL) β = Vons(PART)/c (endMath) /% BIodv(POIo,t) = v/c × E0pcv(POIp,t)) ∇BTodv(POIo,t) = 1/c*[4*π*Jodv(POIo,t) + ∂[∂(t): E0pdv(POIp)]] /* NOTE - I have to check Appendix A later... DIFFERENT from Maxwell equation equivalent! : Doesnt have the curl operation on B, rate of change of E0. /********************************************************** /*------> (4-02) Faradays_Law ref : https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction Maxwell–Faraday equation in integral form by Kelvin-Stokes theorem : integral form, C is a closed curve : /$ ∮[•∂(l),.over.∂Σ: ET) = ∬[•dArea ,.over.: ∂[∂(t): B]] /% ∮[•∂(l),.over.∂Σ: E ) = ∬[•∂(Area),.over.: ∂[∂(t): B]] /*differential form : /$ ∇ET(r,t) = -∂[∂(t): B(r,t)] /% ∇E (r,t) = -∂[∂(t): B(r,t)] /*where : ∂Σ is a surface bounded by the closed contour ∂Σ, Recast integral form using /$ B = ∬[dArea: B(r,t)•n], B(r,t) for B (same for ET) /% B = ∬[∂(Area): B(r,t)•n], B(r,t) for B (same for E) /* Still need to check the primes, etc... - looks similar except for 1/c and lack of a curl operator for E. however, "cgs units" seem to haver pulled a trick in another example??? BUT - important point is the reversal of the integration/derivative operators!!! THIS SEEMS TO BE A HUGE ERROR - local distributions may average to zero, but could have extremely important local effects even if overall they net out? Also - should have dot product of E & dl !! (implied by Lucas for vector multiplication unless specified otherwise?) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn Problem_or_challenge 04_02 Faradays_Law /$ ∫[•∂(l): ET(r´,t´)] = -1/c*∂[∂(t): ∫[∂(Area): B(r,t) •n]] ∫[•∂(l): ET(r´,t´)] = - ∫[∂(Area): ∂[∂(t): B(r,t))]•n] /% ∫[•∂(l): ETodv(POIp(t),t)] = -1/c*∂[∂(t): ∫[∂(Area): BTodv(POIo,t) •n]] ∫[•∂(l): ETodv(POIp(t),t)] = - ∫[∂(Area): ∂[∂(t): BTodv(POIo,t))]•n] /* (OK - but DIFFERENT from Maxwell equation equivalent! : missing curl(E). Does this give good results for experimental data? another important difference is the reversal of the integration/derivative operators!! /********************************************************** /*------> (4-03) Gauss_Electrostatic_Law nh is a unit vector normal to the surface dA https://en.wikipedia.org/wiki/Electrostatics integral form : /$ ∬[•dArea : ET] = 1/ε0*Qenclosed = ∫[•∂^3r,.over.V: ρ/ε0) /% ∬[•∂(Area): E ] = 1/ε0*Qenclosed = ∫[•d^3r,.over.V: ρ/ε0) /* where d^3r = dx*dy*dz differential form via the divergence theorem : /$ ∇•ET = ρ/ε0 /% ∇•E = ρ/ε0 /* Use the integral form, with /$ •∂(Area) replaced by •nh*∂(Area) ET(r,t) for ET 4*π*Q(particle) for ∫[•∂^3r,.over.V: ρ/ε0) /*-> again, which variables have ε0, μ0, π or not, Ill have to check later /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_03 Gauss_Electrostatic_Law /$ ∫[∂(Area): ET(r,t)•nh] = 4*π*Q(particle) /% ∫[∂(Area): E0pdv(POIp)•Roch(POIo)] = 4*π*Q(particle) /* OK - same as conventional expression again, which variables have ε0, μ0, π or not, Ill have to check later /********************************************************** /*------> (4-04) Gauss_Magnetostatic_Law /*# https://en.wikipedia.org/wiki/Magnetostatics Gauss's law for magnetism (steady state!) integral form : /$ ∮[•∂(Area),.over.S: B) = 0 /% ∮[•∂(Area),.over.S: B) = 0 /*differential form : /$ ▽•B = 0 ; /% ▽•B = 0 ; /*reference : Jackson 1999 p179h0.2 /$ ▽•B = 0 ; /% ▽•BTpdv(POIp,t) = ▽•BTodv(POIo,t) = 0 (BIpdv(POIp) = 0 always) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_04 Gauss_Magnetostatic_Law /$ ▽•B = 0 ; /% ▽•B = 0 ; /* (OK - piece of cake... remember - this is for no charge accumulation, steady state electric, magnetic fields!! /********************************************************** /*------> (4-05) Lenz_Induction_Law /* reference : https://en.wikipedia.org/wiki/Lenz%27s_law Wikipedia mentions Lenzs law as being qualitative, relating to direction of Faradays law. But they also say : "... Lenzs law /ˈlɛnts/ is a common way of understanding how electromagnetic circuits obey Newtons third law and the conservation of energy.[1] ..." This concurs with oft-repeated statements by Lucas /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "Question 04_05 Lenz_Induction_Law /% EIodv(POIp(t),t) ∝ -E0odv(POIo,t) = -lambda(Vonv)*E0odv(POIo,t) /* (Qualitatively OK - but I can't seem to find support for this specific form. /********************************************************** /*------> (4-05a) E as sum of E0 & Ei This is very important, especially as Lucas suggests that the static and induced fields behave differently (eg induced fields interact NON-LINEARLY? Ill have to check later. (Hooper & other references from Lucas) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_05a E as sum of E0 & Ei /$ ET(r,v,t) ∝ -E0(r,t) = -λ(v)*E0(r,t) /% ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) /* Didnt look for this yet I havent looked for this yet, but Jackson shows "Normal" linear super´osition, except solid materials, extreme conditions. /********************************************************** /*------> (4-06) Lorentz_Force_Law NOTE - This looks WRONG compared to (4-06), but I think it is (4-06) that is WRONG! 25Aug2015 Lucas04_6 Lorentz_force_law : missing a q in the second term? see p64h0.5 Lucas04_6 versus p69h0.65 Lucas (4-26) - in the latter the charge appears in both terms : Lucas04_26 := Lorentz_force_ /$ F(r - v*t,t) = q*ET (r - v*t,t) + q/c*[vBi(r - v*t,t)] ; /%^% F(ro - vo*t,t) = Q(particle)*E (ro - vo*t,t) + Q(particle)/c*[Vonv(PART)BIodv(ro - vo*t,t)] ; /* Note that in Lucas (4-26) Lucas DERIVES the Lorentz force simply from the Universal force, so he suggests it is NOT a fundamental force. also WRONG? - dividing by c, negative sign rather than positive Note : The "c" in Gaussian units is a persistent source of my errors, but Ill leave it be. reference : Jackson 1999 p557h0.7 Equation (11.144) covariant form of Lorentz force.. This section discusses tensors of 4th order /$ ∂/dτ(p^α) = m*∂/dτ(U^α) = q/c*F^(α*β)*U_b /* wow - not straightforward identifying Lorentz force law in Jackson's book!! reference : https://simple.wikipedia.org/wiki/Lorentz_force /$ F = q*ET + q*vB /%^% F = Q(particle)*E + Q(particle)*Vonv(PART)B /* This agrees with (4-26), but disagrees with (4-06) same issue with : https://in.answers.yahoo.com/question/index;_ylt=AwrTcdYh5.hV8uQAuVUXFwx.;_ylu=X3oDMTBzcnZmYjNuBHNlYwNzcgRwb3MDMTAEY29sbwNncTEEdnRpZAM-?qid=20091114100116AA701h6 reference : http://www.mae.ncsu.edu/buckner/courses/mae535/VanDenBroeck.pdf also has q with B Below I'll add in the (r,v,t) etc add_eqn "Lucas_typo_or_omission 04_06 Lorentz_Force_Law /$ F(r,v,t) = q* ET(r,v,t) - v/cB(r,v,t) /%^% FLENZodv(POIo,t) = Q(particle)*[ ETodv(POIo,t) + Vonv(PART)BTodv(POIo,t) ] /* (WRONG (typo probably)- Lucas is missing the "q" in q*vB(r,v,t)- see (4-06) versus (4-26) and web references ----+ Dimensionality check (see also "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" ... For now, I havent got this to work!!!!!!!! /********************************************************** /*------> (4-07) Galilean_transformation This is straightforward, so I wont check. (eg Jackson 1999 p515h0.55 Eq (11.1) 07Jan2016 Actually - this caused a lot of trouble and wasted work, as earlier assumptions about geometry turned out to be incorrect. A key issue is that the transformation DEPENDS ON the [particle, observer] reference frames to be EXACTLY ewqual at time t=0, after which the particle frame moves with vo*t. My attempts to generalize the geometry (rotation, translation of observer frame, led to hideousy complex expressions. Of course, Jacobians would handle that, but by hiding detail to make it "look simple". In essence, the simple, powerful expressions of physics are only such for braindead system setsups. /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_07 Galilean_transformation /$ Rpcv = r - v*t and t´ = t /%^% Rpcv = ro - vo*t and t´ = t /* OK - straightforward (eg Jackson 1999 p515h0.55 Eq (11.1) /************************* >>>>>>>>> Howell - Appendix A, Derivation of the Biot-Savart and Grassman form of Amperes Law This is in a separate file!! p178 Appendix A : Derivation of the Biot-Savart and Grassman Form of Amperes Law /$LA_01 Fij = i*i´* [ r*(dsi•dsj´)*φ(r) + dsi*(dsj´•r)*P2(r) + dsjp*(dsi•r)*P3(r) + r*(dsi•r)*(dsjp•r)*ψ(r) ] ; /% Fij = i*i´* [ r*(dsi•dsj´)*P(r) + dsi*(dsj´•r)*P2(r) + dsjp*(dsi•r)*P3(r) + r*(dsi•r)*(dsjp•r)*ψ(r) ] ; /* This has to wait for some time before I can get to it. I will work on the front part of the book first. /********************************************** ; >>>>>> 4.2 - Derivation of Electrodynamic Force Law p64h0.9 "... ..." : /********************************************************** /*------> (4-08) Induced_magnetic_flux_density from Amperes law Lucas p64 /$4-01 Bi(r,v,t) = (v/c)E0(r,t) /%^% BIodv(POIo,t) = (Vonv(PART)/c)E0odv(POIo,t) /* The Grassman form of the generalized Ampere force law is based on derivations in Appendix A (eq (A19). (4-08) is the derivation of (4-01) from the Grassman/Biot-Savart form of Amperes Law This is derived in Appendix A... /$ q/c*(vr´)/rs^3 = (v/c)E0(r,t) /%^% Q(particle)/c*(Vonv(PART)Rpcv(POIp))/Rocs(POIo)^3 = (Vonv(PART)/c)E0odv(POIo,t) /* reference : Jackson 1999 p176h0.15 Equation (5.5): /$ B = k*q*(vx )/|x |^3 /%^% B = k*Q(particle)*(Vonv(PART)x )/|x |^3 /* replace x with r, and note that this agrees with the "q/c" part of (4-8) : /$ B = q*k*(vr´)/r´s^3 ; /%^% B = Q(particle)*k*(Vonv(PART)Rpcv(POIp))/Rpcs(POIp)^3 ; /* But should this be Ei, not E0? That would give you : /$ Bi(r,v,t) = q/c*(vr´)/rs^3 = (v/c)E0(r,t) /%^% BIodv(POIo,t) = Q(particle)/c*(Vonv(PART)Rpcv(POIp))/Rocs(POIo)^3 = (Vonv(PART)/c)E0odv(POIo,t) /* Additional issue : p66h0.2 "... Thus it was logically inconsistent to introduce retardation effects into classical electrodynamics for non-rediation situations. ..." For Lucas, maybe, but it IS a legitimate concern!?! /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_08 Induced_magnetic_flux_density from Amperes law /$ Bi(r,v,t) = (v/c)E0(r,t) /% BIodv(POIo,t) = (Vons(PART)/c)E0pdv(POIp) /* OK - Must check derivation of Appendix A Equation (A19) /********************************************************** /*------> (4-09) Frame_transformation_info_lost by Maxwell Lucas - p65h0.85 "... Note that if Amperes law, as represented by equation (4-8), is cast into its usual Maxwell form of equation (4-9), the reference frame transformation information is lost. ..." reference : Jackson 1999 p179h0l.75 Equation (5.21) /$ ∇B = μ0*J(x´) - μ0*/4/π*∇∫[∇´•J(x´)/|x - x´|]*∂^3(x) /% ∇B = μ0*J(x´) - μ0*/4/π*∇∫[∇´•J(x´)/|x - x´|]*d^3(x) /*This is very similar to Lucas expression, but there are a couple of problems : 1. units - I can swap μ0 & c, but will end up with ε0 in there somewhere This looks like an error by Lucas, but also may be due (partially or fully) to unfamiliar unit issues (See "Gaussian vs SI") 2. 3rd derivative of x vs 1st derivative of r (???) Re-express Jackson's (5.20) with r instead of x : /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "Lucas_typo_or_omission 04_09 Frame_transformation_info_lost by Maxwell /$ ∇B(r,v,t) = 4*π/c*J(r,t) + 1/c *∇∫[∂(r): ∇•J(r,t)/|r - r´|} ≈ 4*π/c*J(r,t) /%^% ∇BTodv(POIo,t) = 4*π/c*J(Rocv(POIo),t) + 1/c *∇∫[∂(Rocv(POIo)): ∇•J(Rocv(POIo),t)/|Rocv(POIo) - Rpcv(POIp)|} ≈ 4*π/c*J(Rocv(POIo),t) /* when second term is ignored /$ ∇B = μ0*J(r´) + μ0/4/π*∇∫[∂^3r: ∇•J(r,t)/|r - r´|} ∇B = μ0*J /% ∇BTodv(POIo,t) = 4*π/c*Jodv(POIo,t) + 1/c *∇∫[∇•Jodv(POIp,t)/|Rpcs(POIo(t),t) - Rpcs(POIp)|]∂(r) ∇BTodv(POIo,t) = μ0*Jpdv(POIp,t) /* for steady-state magnetic phenomena! (NOT the same!! is this d^3 a typo in Lucas? Lucas versus Jackson - third derivative d^3r´, my typical problem with [c,μ0,ε0,4,π] NOTE : The second expression is for steady-state magnetic phenomena! |r - r´| term is intriguing => Vons(PART)*t? NOO!!! - special formulation needed /********************************************************** /*------> (4-10) Galilean_transformation Lucas04_10 := Lucas04_07 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_10 Galilean_transformation see Lucas04_07 see Lucas04_07 OK - repeat statement, no need to re-check /********************************************************** /*------> (4-11) E&B_fields_static_plus_induced 04Sep2015 - As I currently understand it, only the E0 versus Ei distinction is critical as the later is non-linear superposition? /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_11 E&B_fields_static_plus_induced /$ ET(r,v,t) = E0(r,t) + Ei(r,v,t) B(r,v,t) = B0(r,t) + Bi(r,v,t) /% ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) BTodv(POIo,t) = B0odv(POIo,t) + BIodv(POIo,t) /* critical distinction between static & induced - have to check later OK - but where is the effect of the critical distinction between static & induced - have to check later where B0odv(POIo,t) = 0 in Chapter 4 /********************************************************** /*------> (4-12) E Galilean transformation particle to observer frames follows from Lucas's notation see (4-07) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_12 E Galilean transformation particle to observer frames /$ ET(r´,t´) = ET(r - v*t,t) /% (ETpdv(POIp)=E0pdv(POIp)) = ETodv(POIo,t) /* (OK, EASY - key point, I need to research results and opinions, seems correct /********************************************************** /*------> (4-13) Total B magnetic flux density as induced from E0 + Ei /*$ cat >>"$p_augmented" "$d_augment""04_13 work.txt" /*_endCmd /********************************************************** /*------> (4-14) B&E point charge - substituted Amperes law /*$ cat >>"$p_augmented" "$d_augment""04_14 work.txt" /*_endCmd /********************************************************** /*------> (4-14a) Point particle and symmetry This is stated based on symmetry of field around a single moving charge, to which the inuced field is attached. Also rh´=r´/|r´|, ie. unit vector in direction of r All this does is convert a scalar (magnitude of Ei) to a vector, but why was a scalar to begin with? /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_14a Point particle and symmetry /$ Ei(r´,t´) = |Ei(r´,t´)|*r´/|r´| Ei(r´,t) = |Ei(r´,t´)|*r´/|r´| /%^% EIodv(Rpcv(POIp(t),t) = |EIodv(Rpcv(POIp(t),t)|*Rpcv(POIp)/|Rpcv(POIp)| (EIpdv(POIp)=0) = (EIpds(POIp)=0)*Rpch(POIo(t),t) /* OK, simple /********************************************************** /*------> (4-15) E,B for symmetry point charge @v_const Adapted non-[derivative, integral] form of Amperes law /*$ cat >>"$p_augmented" "$d_augment""04_15 work.txt" /*_endCmd /********************************************************** /*------> (4-16) E,B for symmetry point charge @v_const /*$ cat >>"$p_augmented" "$d_augment""04_16 work.txt" /*_endCmd /********************************************************** /*------> (4-17) Spherical coordinate transforms Howells clarifications - Lucas converts to spherical coordinates see Howell - Symbols for Bill Lucas's book "The Universal Force, vol1" Special figure for this Expression for rps(Rocs,Oo,vo,t) From Diagram : /$ |r| = Rocs = [ (Rocs*sin(θ))^2 + (Rocs*cos(θ))^2 ]^(1/2) 1) |r´| = r´s = [ (r´s*sin(θ´))^2 + (r´s*cos(θ´))^2 ]^(1/2) /* Noting that : /$2) r´s *sin(θ´) = Rocs*sin(θ) 3) Rocs*cos(θ) = vs*t + r´s*cos(θ´) /* Substituting 2) & 3) into 1) : /$4-17a) r´s = [ (Rocs*sin(θ))^2 + (Rocs*cos(θ) - vs*t)^2 ]^(1/2) /* Expression for vov X r´ /$5) v X r´ = v X (r - v*t) = v X r - v X v*t but v X v = 0 (collinear) so : 6) v X r´ = v X r = |v|*|r|*sin(θ)*φ´hat 4-17b) v X r´ = vs*Rocs*sin(θ)*φ´hat /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "Re_check_later 04_17rev2 Spherical coordinate transforms /$ |r´| = |r - v*t| = sqrt((Rocs*sin(θ))^2 + (Rocs*cos(θ) - vs*t)^2) vr´ = v(r - v*t) = vr = vs*Rocs*sin(θ)*φ´hat /% Rpcs(POIp) = { [Rocs(POIo,t)*sin(Aθoc(POIo))]^2 + [Rocs(POIo,t)*cos(Aθoc(POIo) - Vons(PART)*t]^2 }^(1/2) ????Vonv(PART) X Rpcv(POIp) = Vons(PART)*Rocs(POIo,t)*sin(Aθoc(POIo))*APph ???? *Rodh(Vonv_X_Rpcv(POIo)) /* ( 02Jan2016 OK - easy BUT, I still need to check angle basis below... Recheck my HFLN expression! /********************************************************** /*------> (4-18) Changing magnetic flux linked by a circuit proportional to induced E field around the circuit from Jackson 1999, p211h0.45 Eqn (5.141) /$ ∮[•d(l´),.over.C: ET) = -1/c*∂[∂t: ∮[dArea,.over.S: (B•n)) /% ∮[•d(l´),.over.C: E) = -1/c*∂[∂t: ∮[∂(Area),.over.S: (B•n)) /*where : nh is a unit vector normal to the surface S at the point of integration from Jackson to Lucas notation - n = nh - B = Bi(ro - vo*t,t) (observer frame), - E = Eip(r´,t´) (particle/system frame) /$ ∮[•d(l´),.over.C: Ei´(r´,t´)) = -1/c*∂[∂t: ∮[dArea,.over.S: (Bi(r - v*t,t)•nh)) /%^% ∮[•d(l´),.over.C: Eip(Rpcv(POIp),t´)) = -1/c*∂[∂t: ∮[dArea,.over.S: (BIodv(ro - vo*t,t)•Roch(POIo))) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "Question 04_18 Faradays law -> B linked by circuit = induced E around circuit /$ ∮[•d(l´),.over.C: Ei´(r´,t´)) = -1/c*∂[∂t: ∮[dAreap,.over.S: Bi(r - v*t,t)•nh) ∮[•d(l´),.over.C: Ei´(r´,t´)) = - *∂[∂t: ∫[dArea,.over.S: Bi(r - v*t,t)•nh) /%^% ∮[•d(l´),.over.C: (EIpdv(POIp) = 0)) = -1/c*∂[∂t: ∮[dArea,.over.S: (BIodv(POIo,t) •Roch(POIo))) ∮[•d(l´),.over.C: Eip(Rpcv(POIp),t´)) = - *∂[∂t: ∫[dArea, .over.S: BIodv(ro - vo*t,t)•Roch(POIo)) /* Perfect - simple formula translation, BUT is Ep = E, as it looks to me that these are used differently/simultaneously p68&69, but it is NOT explained! What is it? /********************************************************** /*------> (4-19) E,B for symmetry point charge @v_const - Stokes theorem /*$ cat >>"$p_augmented" "$d_augment""04_19 work.txt" /*_endCmd /********************************************************** /*------> (4-20) Convective_derivative Jackson1999 p210h0.9 footnote "*" convective derivative : /$ ∂/∂(t) = ∂[∂t: + v•∇ /* where nabla ∇p = gradient in particle refFrame = Cartesian : d(dx : ?)*xvh + ... Spherical : .... /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_20 Convective_derivative /$ ∂/∂(t) = ∂[∂t: + v•∇p] /% d/∂(t) = ∂[∂t: + v•∇p] /* OK - straight defn. Usage explained by Jackson (to generalize Faradays Law?) HOWEVER - I should have been using convective derivatives elsewhere!! - eg as I did with dp[dt : E] in Background (eg file "Howell - Background math for Lucas Universal Force, Chapter 4.odt") /********************************************************** /*------> (4-21) convective derivative of Total magnetic flux density Bi /*$ cat >>"$p_augmented" "$d_augment""04_21 work.txt" /*_endCmd /********************************************************** /*------> (4-22) Kelvin-Stokes integration of convective derivative of Bi total p69h0.2 apply Kelvin-Stokes theorem to Lucas04_21 : gives Lucas04_22 https://en.wikipedia.org/wiki/Kelvin%E2%80%93Stokes_theorem Let γ: [a, b] → R2 be a Piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides R2 into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t))[note 1] and F is a smooth vector field on R3, then: /$ ∮(dΓ: F) = ∬[∂(Area): ∇F) /* Note 1 : γ and Γ are both loops, however, Γ is not necessarily a Jordan curve STRANGE : I dont see the point of deriving (4-22) at all!?? I have simply applied Kelvin-Stokes to (4-19) in the next section (4-23) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "Howell_incomplete 04_22 (Kelvin-Stokes integration of convective derivative of Bi total Lucas ?? - Integral form of Faradays_Law_for_moving_circuit - magnetic fields??? /$ 1/c*∮*∂(Area)´(∇´[Bi(r - v*t,t)v]•np) = 1/c*∮•d(l´)([Bi(r - v*t,t)v]) /* STRANGE : I dont see the point of deriving 4-22 at all!?? didnt do - I dont see the point of deriving 4-22 !?! I have simply applied Kelvin-Stokes to 4-19 in the next section 4-23 /********************************************************** /*------> (4-23) Faradays_Law_for_rest_circuit integral form E,B /*$ cat >>"$p_augmented" "$d_augment""04_23 work.txt" /*_endCmd /********************************************************** /*------> (4-24) E&B for [Faradays + part/obs frameTrans] - towards FU_Faradays_Law /*$ cat >>"$p_augmented" "$d_augment""04_24 work.txt" /*_endCmd /********************************************************** /*------> (4-25) E&B for [Faradays + part/obs frameTrans] - towards FU_Faradays_Law Lucas development Derivation of the Lorentz Force /$ Fp(r´,t´) = F0(r´,t´) + Fi(r´,t´) = q*E0(r´,t´) + q*Ei(r´,t´) = q*E0(r - v*t,t) + q*Ei(r - v*t,t) + q/c*[vBi(r - v*t,t)] /* 1. start with /$ Fp(r´,t´) = F0(r´,t´) + Fi(r´,t´) = q*E0(r´,t´) + q*Ei(r´,t´) /* This is nice, but F = q*E is NOT stated in the original axioms, even though it is basic, so perhaps assumed. Utilising (4-24), plus putting in the observer frame (r´,t´) -> (ro - vo*t,t) (note the change from Ei(r´,t´) to Ei(r´,v´,t´) ) /$ Fp(r´,t´) = q*E0(r - v*t,t´) + q*( Ei(r - v*t,t) + v/cBi(r - v*t,t)) /%^ Fp(Rpcv(POIp),t´) = Q(particle)*E0odv(Rpcv(POIp),t´) + Q(particle)*( EIodv(POIp(t),t) + Vonv(PART)/cBIodv(POIo,t)) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_25 Derivation of the Lorentz Force /$ Fp(r´,t´) = q*E0(r - v*t,t) + q*Ei(r - v*t,t) + q/c*[vBi(r - v*t,t)] /% FTpdv(POIo(p),t) = Q(particle)*{ E0odv(POIo,t) + EIodv(POIo,t) + Vons(PART)/cBIodv(POIo,t) } /* OK - straightforward from 4-24 12Sep2015 - I still have problems with 4-24 /********************************************************** /*------> (4-26) Derived Lorentz Force F_L(const v : q,E,Bi) starting with (4-25) 04_25 Derivation of the Lorentz Force /% FTpdv(POIo,t) = Q(particle)*{ E0odv(POIo,t) + EIodv(POIo,t) + Vons(PART)/cBIodv(POIo,t) } /* from (4-5a) given that /$ E0(r - v*t,t) + q*Ei(r - v*t,t) Fp(r´,t´) = q*ET(r - v*t,t) + q/c*[vBi(r - v*t,t)] /* p69h0.8 "... Note that the Lorentz force law has been derived from Galilean invariance and the experimental fact that the fields are a physical extension of the charge making the electromagnetic force a contact type force! The Lorentz Force law should no longer be considered a fundamental axiom of electrodynamics. ..." 25Aug2015 Howell - This is an important clarification of what Lucas means by the expression "contact force". I have not considered "fields" to be a contact force at all, and others may be confused by this terminology as well. Perhaps this is to contrast his field-based approach to the boson "force particles" of modern physics? from http://particleadventure.org/fermibos.html A fermion is any particle that has an odd half-integer (like 1/2, 3/2, and so forth) spin. Quarks and leptons, as well as most composite particles, like protons and neutrons, are fermions. Bosons are those particles which have an integer spin (0, 1, 2...). All the force carrier particles are bosons, as are those composite particles with an even number of fermion particles (like mesons) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_26 Derived Lorentz Force, F_L(const v : q,E,Bi) /$ F (r´,t´) = q*ET(r - v*t,t) + q/c*[vBi(r - v*t,t)] /% F_LORENTZpdv(POIo,t) = Q(particle)*{ EIodv(POIo,t) + Vons(PART)/cBIodv(POIo,t) } /* OK - very straightforward /********************************************************** /*------> (4-27) Lorentz Force /*$ cat >>"$p_augmented" "$d_augment""04_27 work.txt" /*_endCmd /********************************************************** /*------> (4-28a) Faradays_law_spherical_coords - ∇´Ei(ro - vo*t,t) term /*$ cat >>"$p_augmented" "$d_augment""04_28a work.txt" /*_endCmd /********************************************************** /*------> (4-28b) Faradays_law_spherical_coords - -1/c*dp[dt : Bi(ro - vo*t,t)] term 25Sep2015 revision 1 started (trivial changes) 1. Use 04_16 E,B for symmetry point charge @v_const Derivative form of Amperes law /$L 1/c*∂[∂(t): Bi(r - v*t,t)] = vs/c*rs*sin(θ)*φ´*[3*q*vs/c*(rs*cos(θ) - vs*t)/|rs - vs*t|^5 + 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ] /$H 1/c*∂[∂(t): Bi(r - v*t,t)] = vs/c*rs*sin(θ)*φ´*[3*q*vs/c*(rs*cos(θ) - vs*t)/|rs - vs*t|^5 + 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ] /* Simple! used 4-16, can insert minus signs to fit Lucas format /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_28brev1 Faradays_law_spherical_coordinates - full form /$L ∇´Ei(r - v*t,t) = -1/c*∂[∂(t): B(r´,t)] = -vs/c*rs*sin(θ)*φ´*[ 3*q*vs/c*(rs*cos(θ) - vs*t)/|r - v*t|^5 + 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ] /$H ∇´Ei(r - v*t,t) = -1/c*∂[∂(t): B(r´,t)] = ??? vs/c*rs*sin(θ)*φ´*[ 3*q*vs/c*(rs*cos(θ) - vs*t)/|r - v*t|^5 + 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ] /% ∇´Ei(ro - vo*t,t) = -1/c*∂[∂(t): BIodv(POIo,t)] = -Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))*RAPpdh *{ 3*Q(particle)*Vons(PART)/c*(Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t)/Rpcs(POIo(t),t)^5 + 1/Rpcs(POIo(t),t)/c*∂[∂(t): EIods(POIo,t)] } /* OK - Simple! I used 4-16. 29May2016 - still a concern => But - shouldnt all angles be primed to get particle/system refFrame? /********************************************************** /*------> (4-29a) Faradays_law_spherical_coords - 1st term /*$ cat >>"$p_augmented" "$d_augment""04_29a work.txt" /*_endCmd /********************************************************** /*------> (4-29b) Faradays_law_spherical_coords - 2nd term 25Sep2015 revision 1 started, 27May2016 added quick explanation From the verification of (4-29a) above : By symmetry, the following partial derivatives are zero (with respect to the scalar result!!): /$ ∂[∂(φ): Ei(r´,t)•θ´hat) ∂[∂(φ): Ei(r´,t)•r´h) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn 04_29brev1 : no_issue Faradays_law_spherical_coordinates - dropping term /$ 1/r/sin(θ)*∂[∂Pp: Ei(r - v*t,t)*θ´hat] = 0 1/r/sin(θ)*∂[∂Pp: Ei(r - v*t,t)*θ´hat] = 0 /* OK - easy after fixing (4-29a) above /% 1/Rocs(POIo)/sin(Aθoc(POIo))*∂[∂(φ´): EIods(POIo,t)*Rθpch] = 0 /********************************************************** /*------> (4-30) Faradays_law integrated over θ 24Sep2015 - With this revision, I will take extra care with variable notations (hat, prime, scalar, etc). 25Sep2015 revamped!! Lucas's result looks wrong? Faradays_law_integrated /*$ cat >>"$p_augmented" "$d_augment""04_30 work.txt" /*_endCmd /********************************************************** /*------> (4-31) From Lenzs law and symmetry of local forces Lucas - From Lenzs law and symmetry of local forces, /$ Eis(r - v*t,t)|@O=0 /* should oppose the induced field /$ Eis(r - v*t,t)r´/|r´| /* which is proportional to the moving static field /$ E0s(r - v*t,t)r´/|r´|´ /* OK with concerns - Looks reasonable (see caveats below) and straightforward. No details needed here. /*++++++++++++++++++++++++++++++++++++++ /*add_eqn 04_31 : uncomfortable with this simplification F therefore E balance from Lenzs law and symmetry of local forces /$L Eis(r - v*t,t)|(O=0)*r´h = -λ(v)*E0s(r - v*t,t)*r´h /$H Eis(r - v*t,t)|(O=0)*r´h = -λ(v)*E0s(r - v*t,t)*r´h /* OK with concerns - Looks reasonable (see caveats below) and straightforward, but doesnt Lenzs Law apply to other angles as well? No details needed here. Lenzs Law seems to be a very general, only referring to a proportionality between E0 and EI, but Lucas is not providing any other functional relations. Here Lucas has taken it quite literally!? I do not need to do work here, but I am a bit uncomfortable with this simplification, which has huge implications later /% EIods(POIo,t,Aθpc(POIp) = 0)*Rpch(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo,t)*Rpch(POIo(t),t) /* Notice that the static component E0 isn't direction dependent, just distance The above relation also implies that /% EIods(POIo,t,Aθpc(POIp) = 0) = -λ(Vons(PART))*E0ods(POIo,t) /********************************************************** /*------> (4-31a) Machs principle - Lenz works, SRT & covariant Maxwell fail Lucas makes statement that the use of Lenzs law satifies Machs principle, whereas neither Einsteins Special Relativity Theory (SRT) nor the covariant form of Maxwells equations do. p70h0. /*+++++++++++ /*add_eqn 04_31a "Howell_incomplete Machs principle - Lenz works, SRT & covariant Maxwell fail Provided as statement only not done yet - Important issue for Lucas - I"ll have to think this over... Machs principle - Important issue for Lucas. not done yet. I"ll have to think this over... /********************************************************** /*------> (4-32) EIods(POIo,t=0,1st stage), F therefore E balance - iteration #1 on (4-30) initial - 15Sep2015, rev1 - ??, rev2 - 25Sep2015 15Sep2015 p71h0.33 Iterate [(4-31) into (4-30)] to obtain solution. 09Jun2016rev5 use HFLN, corrected (I hope) [reference frames, notations, derivatives, integrals] 14Jun2016rev6 fix my error (cosOp - 1) !! 22Aug2019 remove (cosOp - 1) term 30Sep2019 corrections, cleanup /*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_32 work.txt" /*_endCmd /********************************************************** /*------> (4-33) EIods(POIo,t=0, 2nd stage), F therefore E balance 29May2016 I had mistakenly used Ei in place of Eis, so this has been corrected in the final result. 14Jun2016rev3 fix my error (cosOp - 1) !! Lucas's version of (4-32) (4-32rev3 F therefore E balance - simplified (4-30) 22Aug2019 start revision used revamped (4-32), finished 26Aug2019 03Sep2019 fix error with Equation (5) below - dropped a term, K3 was incorrect then simply dropped from expressions 24Sep2019 drop K1, add "Highly restrictive conditions" 30Sep2019 correct + sign for K1 /*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_33 work.txt" /*_endCmd /********************************************************** /*------> (4-34) EIods(POIo,t=0, 2nd stage), K_2nd from taking partial derivatives wrt time for previous versions, see "Howell - Old math of Lucas Universal Force.ndf" 16Sep2015 1st or 2nd?, 20Sep2015 rev2, 25Sep2015 rev4 29May2016rev5 14May2016rev6 13Aug2019 !!! ∂[∂(x): sin(x)] ​= cos(x) !!! ????????????????????????? /% 27Aug2019 revamp, hopefully fixing wrong [+,-] issues for (4-37) There is an ambiguity in my expressions with [t, t=0] - sometimes I've left in the wrong form 03Sep2019 fix error in K2 term from 04_33 (actually - this was not a problem?) 03Sep2019 Note that I corrected the results there, which had Aθpc(POIp(t),t), which is an illegal symbol!!! this was corrected to Aθoc(POIp(t),t) or Aθpc(POIo(t),t) 03Sep2019 no net change to end result of 04_34 24Sep2019 following changes to 4-33, drop K1, add "Highly restrictive conditions" 02Oct2019 fix ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a], add K_1st term 14Oct2019 fix generative form for 4-34 (forgot to do that earlier) /#_file_insert "cos - 1 yes, iterative, non-feedback/04_34 work.txt" /*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_34 work.txt" /*_endCmd /********************************************************** /*------> (4-35) E0ods(POIo,t) truncated expression with ONLY E0ods(POIo,t) terms 16Sep2015 1st version, ?17Sep2015? rev1, 21Sep2015 rev2, 26Sep2015 rev3 for previous versions, see "Howell - Old math of Lucas Universal Force.ndf" ??? 04Jan2016 Check if should be redone with "Howell - Key math info & derivations for Lucas Universal Force.odt" 29Aug2019 re-derive based on recent work from (4-32) to (4-34) 03Sep2019 fix ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIp(t),t)) ] : I had sin(Aθpc(POIo(t),t=0))^1/1, should have been sin(Aθpc(POIo(t),t=0))^2/2 24Sep2019 following changes to 4-33, drop K1, add "Highly restrictive conditions" 02Oct2019 percolate fix to ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a], note K_2nd term /#_file_insert "cos - 1 yes, iterative, non-feedback/04_35 work.txt" /*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_35 work.txt" /*_endCmd /********************************************************** /*------> (4-36) ETods(POIo,t) expression with ONLY ETods(POIo,t) terms 17Sep2015 2nd iteration 03Sep2019 Redone with HFLN - it now works!!!!! /*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_36 work.txt" /*_endCmd ********************************************************** /*------> (4-37) Er and the binomial series, leading to the relativistic correction factor 27Sep2019 - did not finish - revamp in next attempt with updated previous equations 4-[32 to 36] 27Sep2019 using updated previous equations 4-[32 to 36] 02Oct2019 add [K_1st, K_2nd] terms after fixing ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a] fix ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a] 11Oct2019 use corrected form of EIods(POIo,t,2nd stage) with "+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}" Approach for 22Aug2019, 03Sep2019 after correcting recent errors in 04_35 : 1. Don't track Lucas too closely - he has wrong coefficients 2. Iterations pass through same operations as (4-32) through (4-36) 3. Howells "dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)] - see proper E0odv(POIo,t) vector approach" from "Howell - Background math for Lucas Universal Force, Chapter 4.odt" I was very lazy with symbols -0 many improper uses!!! Must clean up thre t=0 notational mess for [integrals, derivatives] /*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_37 work.txt" /*_endCmd /********************************************************** /*------> (4-38) Binomial_expansion_for_E0_terms 23Sep2015 start 1st attempt /*$ cat >>"$p_augmented" "$d_augment""04_38 work.txt" /*_endCmd /********************************************************** /*------> (4-39) E(r,v) for constant velocity, non-point charge, observer reference frame /*$ cat >>"$p_augmented" "$d_augment""04_39 work.txt" /*_endCmd /********************************************************** /*------> (4-40) Gauss_Electrostatic_Law /* Lucas - take from Jackson1999 p27 /$ ∬[∂(Area): ET(r)nh) = 4*π*q /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_40 L(v) expression for Gauss law for electric charge /$L 4*π*q = ∬[∂(Area): ET(r)nh) /% 4*π*Q(particle) = ∬[∂(Area)´: ETodv(POIo,t)RNpch) /* OK - no need to do as it can be found from Jackson1999, standard formula /********************************************************** /*------> (4-41) L(v) expression for Gauss law for electric charge /*$ cat >>"$p_augmented" "$d_augment""04_41 work.txt" /*_endCmd /********************************************************** /*------> (4-42) Special integral with binomial series (1 - b^2*sin^2(O))^(3/2) /* Started ?Sep or Oct2015? /*$ cat >>"$p_augmented" "$d_augment""04_42 work.txt" /*_endCmd /********************************************************** /*------> (4_43) E&B_fields_self_consistent Lucas /$ ET(r,v) = E0(r) + Ei(r,v) = E0(r)*(1 - β^2)/(1 - β^2*sin(θ)^2)^(3/2) Bi(r,v) = v/cET(r,v) /* Howell - from (4-42) /$ λ(v) = β^2 /* Substituting into (4-39) 04_39 E(r,v) for constant velocity, non-point charge, observer reference frame /$ ET(r,v) = (1 - λ(v))*E0(r)/(1 - β^2*sin(θ)^2)^(3/2) ET(r,v) = E0(r)*(1 - β^2)/(1 - β^2*sin(θ)^2)^(3/2) /* From (4-13) 04_13 Total B magnetic flux density as induced from E0 + Ei /$ B(r´,t´) = (v/c)[ E0(r´,t´) + Ei(r´,t´) ] B(r´,t´) = (v/c)ET(r´,t´) /* 29May2016 - As E&B are measured at a POI they give the same result in either RFp or RFo for that POI. What does matter is whether we are refering to a fixed POIo, or POIp. There is no B in the latter case. earlier comment : Lucas drops the primes, so I will too. Also - we are dealing with a constant v and particle/system frame of reference, and (from 04_13 : ***where B0 is dropped as it "... is not electrostatic in nature ..." (p67h0.4) /$ B(r,v,t) = B0(r,t) + Bi(r,v,t) = Bi(r,v,t) /* so /$ B(r´,t´) = (v/c)ET(r´,t´) /* Lucas does NOT substitute the expression (4-39) for E(r,v) AND - he includes the ELECTROSTATIC E???? /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_43 E&B_fields_self_consistent /$L ET(r,v) = E0(r)*(1 - β^2)/(1 - β^2*sin(θ)^2)^(3/2) Bi(r,v) = (v/c)ET(r,v) /$H /% ETodv(POIo,t) = E0odv(POIo,t)*(1 - β^2)/(1 - β^2*sin^2(Aθpc(POIo(t),t)))^(3/2) BIodv(POIo,t) = Vons(PART)/cEIodv(POIo,t) /* OK - simple, although I think primes are needed to denote particle reference frame (RFp) for angle theta (Op). This does not apply for E,B vectors, for which the primes are unimportant - the direction is the same in both (RFo) and (Rfp). /********************************************************** /*------> (4-44) F_total by moving charge distribution on a test charge q' /*$ cat >>"$p_augmented" "$d_augment""04_44 work.txt" /*_endCmd /********************************************************** /*------> (4-45) Vector identities for Lorentz Force derivation Lucas - "The following identities were used for Lucas04_44 " Howell - checks from references Lucas's expression (4-45) follows directly from (4-46) /$ v/c[v/cE0(r,v)] = (v/c)•E0(r)*(v/c) - (vs/c)^2*E0(r) = (v/c)•E0(r)*[(v•rh)*rh/c - rh(rhv)/c] - (vs/c)^2*E0(r) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_45 Vector identities for Lorentz Force derivation /$L v/c[v/cE0(r,v)] = (v/c)•E0(r)*[(v•rh)*rh/c - rh(rhv)/c] - (vs/c)^2*E0(r) /$H ??? /% Vonv(PART)/c[Vonv(PART)/cE0(r,v)] = (Vonv(PART)/c)•E0odv(POIo,t)*[(Vonv(PART)•Roch(POIo))*Roch(POIo)/c - Roch(POIo)(Roch(POIo)Vonv(PART))/c] - (Vons(PART)/c)^2*E0odv(POIo,t) /* OK - not required as follows directly from (4-46) should Roch(POIo) below be Rodh(Vonv_X_Rpcv(POIo)) from file "Howell - Background math for Lucas Universal Force, Chapter 4.odt"? /********************************************************** /*------> (4-46) Vector_operations used for the Lorentz force Lucas - "The following identities were used for Lucas04_44 " Howell - checks from references From Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt scalar triple products Kreyszig Section 5.9 p213-216 /$1) β(c∂) = (β∂)*c - (βc)*∂ /* This takes care of the first expression The second equation in (4-46) : /$2) v = v - (vrh)*rh + (vrh)*rh = (vrh)*rh - rh(rhv) /* The first step is straightforward - just add and subtract the same term : /$ v = v - (vrh)*rh + (vr)*r /* Reverse-processing the 2nd step, using (1) /$ rh(rhv) = (rhv)*rh - (rhrh)*v /* but r_hr_h = 1 (unit vectors same direction), therefore /$2a) rh(rhv) = (rhv)*rh - v /* Putting (2a) into (2) /$ v = (vrh)*rh - rh(rhv) = (vrh)*rh - ((rhv)*rh - v) = v /*So this works /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 04_46 Vector_operations used for the Lorentz force /$L A(BC) = (A•C)*B - (A•B)*C /$H A(BC) = (A•C)*B - (A•B)*C /* OK - very simple, from textbooks /********************************************** ; >>> Lucas 5 - Extension of the Universal Force Law to include acceleration /********************************************** ; >>>>>> 5.1 - Generalized electromagnetic potential U(r,v) /********************************************************** /*------> 05_01 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_01 ?title? same as Lucas04_44 no need - same as Lucas04_44 no need - same as Lucas04_44 /********************************************************** /*------> Lucas Generalized_potential_U /$ U(r,v) = -F(r,v)•r ∂[∂t: U) = -F(r,v)•∂[∂t: r) - ∂[∂t: F(r,v))•r = -F(r,v)•v - ∂[∂t: F(r,v))•r /* For stability iunder a constant force : /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_02 Generalized_potential_U /$ ∂[∂t: U) = -F(r,v)•v - ∂[∂t: F(r,v))•r /* havent done yet /* havent done yet /********************************************************** /*------> 05_03 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_03 ?title? /$ ∂[∂t: U(r,v)) = -F(r,v)•v /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Generalized_potential_U /$ U(r,v) = q*q´/r*(1 - β^2)/(1 - β^2*sin(θ)^2)^(1/2) = q*q´/r*(1 - β^2)/[r^2 - {r(rβ)/r^2}]^(1/2) where β=v/c and ∂[∂t: U(r,v) = - v•F(r,v,a) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_04 Generalized_potential_U /$ U(r,v) = q*q´/r*(1 - β^2)/[r^2 - {r(rβ)/r^2}]^(1/2) /* where /$ β=v/c /* and /$ ∂[∂t: U(r,v) = - v•F(r,v,a) /* havent done yet /********************************************************** /*------> Lucas /$ v•F = -∂[∂t: U(r,v)) = -∂/∂(t)[q*q´*(1 - β^2)/[r^2 - {r(rβ)}^2/r^2]^(1/2)] = -q*q´*{ ∂/∂(t)[(1 - β^2)/[r^2 - {r(rβ)}^2/r^2]^(1/2) + (1 - β^2)/[r^2 - {r(rβ)}^2/r^2]^(1/2) * (-1/2)*∂/∂(t)[r^2 - {r(rβ)}^2/r^2]^(1/2) / [r^2 - {r(rβ)}^2/r^2]^(3/2) } = q*q´*{ 2*(v/c)•(a/c)/[r^2 - {r(rβ)}^2/r^2]^(1/2) + 1/2*(1 - β^2)/[2*r•v - {r(rβ)}^2*2*r•v/r^4] / [r^2 - {r(rβ)}^2/r^2]^(3/2) } + q*q´*{ 1/2*(1 - β^2)*2*{r(rβ)}/r^2 • ∂/∂(t){r*(r•β) - β*(r•r)} / [r^2 - {r(rβ)}^2/r^2]^(3/2) } = q*q´*{ 2*(v/c)•(a/c)/[r^2 - {r(rβ)}^2/r^2]^(1/2) + (1 - β^2)*r•v* [1 - {r(rβ)}^2/r^4] / [r^2 - {r(rβ)}^2/r^2]^(3/2) } + q*q´ *(1 - β^2) *{r(rβ)} /r^2 •{v*(r•β) + r*v^2/c + r*(r•a/c) - a/c*r^2 - 2*β*(v•r)} / [r^2 - {r(rβ)}^2/r^2]^(3/2) } = q*q´/r^2*{ (1 - β^2)*v•r + 2*r^2/c^2*v•a } / [r^2 - {r(rβ)}^2/r^2]^(1/2) + q*q´ *(1 - β^2) *??x?? *{r(rβ)} •{-v/c*(v•r) + r*v^2/c} + {r*(r•β) - β*r^2}•r(ra/c) } / [r^2 - {r(rβ)}^2/r^2]^(3/2) = q*q´/r^2*{ (1 - β^2)*v•r + 2*r^2/c^2*v•a } / [r^2 - {r(rβ)}^2/r^2]^(1/2) - q*q´ *(1 - β^2)* { r(rβ)•v/c*(v•r) + r^2*r(ra/c)•β } / [r^2 - {r(rβ)}^2/r^2]^(3/2) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_05_pre ?title? /$ ( v•F = q*q´/r^2*{ (1 - β^2)*v•r + 2*r^2/c^2*v•a } / [r^2 - {r(rβ)}^2/r^2]^(1/2) - q*q´ *(1 - β^2)* { r(rβ)•v/c*(v•r) + r^2*r(ra/c)•β } / [r^2 - {r(rβ)}^2/r^2]^(3/2) ) /* havent done yet /* havent done yet /********************************************************** /*------> p79h0.2 the following vector identities were used /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_06 ?title? /$ A(BC) - (A•C)*B - (A•B)*C /$L r•r(rβ) = (r•β)*(r•r) - (r•r)*(r•β) = 0 /$H??? r•r(ra) = (r•a)*(r•r) - (r•r)*(r•a) = 0 /* havent done yet havent done yet /********************************************************** /*------> 28Aug2015 - look at this later. Might help to get rid of "extra" "r"!?!? H5_07 := Universal_ED_force_with_acceleration_simplified := /$ F(r,v,a) = = q*q´/r^2*(1 - β^2)*r + 2*r^2/c^2*a } / [r^2 - {r(rβ)}^2/r^2]^(1/2) - (1 - β^2)* ???? do later ???? { (β•r)*r(rβ) + (r•r)*r(ra/c^2) } / [r^2 - {r(rβ)}^2/r^2]^(3/2) } /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_07 Universal_ED_force_with_acceleration /$L ( F(r,v,a) = = q*q´/r^2* [ + { (1 - β^2)*r + 2*r^2/c^2*a } / [r^2 - {r(rβ)}^2/r^2]^(1/2) - (1 - β^2)* { (β•r)*r(rβ) + (r•r)*r(ra/c^2) } / [r^2 - {r(rβ)}^2/r^2]^(3/2) ] ) /* havent done yet havent done yet /********************************************************** /*------> 05_08 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_08 Phipps_ED_force_relativistic_transverse Wesleys_ED_force_relativistic_circular /$ F(r,v,a: rperpendicular to v,a = 0) = q*q´*rh/r^2*[(1 - β^2)/(1 - β^2*sin(θ)^2)^(1/2) ](@O=π/2) ≈ q*q´*rh/r^2* (1 - β^2)^(1/2) /* havent done yet havent done yet /********************************************************** /*------> p80h0.8 In the non-relativistic limit v< Lucas F_Lienard_Wichert_electric_field_v_muchless_c /$ ≈ q*E_v_muchless_c ≈ q*E0 + q*(Ea + Erad) where q*(Ea + Erad) ≈ q*q´/c^2/r*[2*a + r*(r•a) - a] therefore F_Lienard_Wichert_v_muchless_c = q*q´*r/rs^3 + q*q´/c^2/r*[2*a + r*(r•a) - a] /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_10 /$ F_Lienard_Wichert_electric_field_v_muchless_c ( F_Lienard_Wichert v<>>>>> 5.2 - Acceleration fields and radiation ; /********************************************************** /*------> 05_11 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_11 Acceleration fields and radiation /$ FE = q*ET(r) = q*[E0(r) + Ei(r) ] FM = v/c[B0(r) + Bi(r)] /* havent done yet /* havent done yet /********************************************************** /*------> 05_12 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_12 Acceleration fields and radiation /$ Ei(r) = Ev(r) + Ea(r) + Erad(r) + ... Bi(r) = Bv(r) + Ba(r) + Brad(r) + ... /* havent done yet /* havent done yet /********************************************************** /*------> 05_13 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_13 Acceleration fields and radiation /$ F = F_E0 + F_Ev + F_Ea + ... + F_B0 + F_Bv + F_Ba + ... + F_rad /* havent done yet /* havent done yet /********************************************************** /*------> Lucas /$ F_E0 + F_Ev = q*q´*r/rs^3*(1 - β^2) /(1 - β^2*sin(θ)^2)^(1/2) = q´*(E0 + Ev) F_B0 + F_Bv = = q*q´*r/rs^3*(1 - β^2)*β^2*sin(θ)^2 /(1 - β^2*sin(θ)^2)^(1/2) + q*q´ /rs^4*(1 - β^2)*(β•r)*r(rβ)/(1 - β^2*sin(θ)^2)^(3/2) /* /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_14 Acceleration fields and radiation /$ F_E0 + F_Ev = q´*(E0 + Ev) F_B0 + F_Bv = = q*q´*r/rs^3*(1 - β^2)* β^2*sin(θ)^2 /(1 - β^2*sin(θ)^2)^(1/2) + q*q´ /rs^4*(1 - β^2)*(β•r)*r(rβ)/(1 - β^2*sin(θ)^2)^(3/2) /* havent done yet /* havent done yet /********************************************************** /*------> Poynting vector S - radiation from charged particle acceleration Poynting_radiation_flux_from_charged_particle_acceleration /$ S = c/4/π*|Erad|^2*r = q^2/4/π/c^3*|r(ra)|^2/r^2*(1 - β^2)^2/(1 - β^2*sin(θ)^2)^3 /* /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_15 Poynting_radiation_flux_from_charged_particle_acceleration /$ S = q^2/4/π/c^3*|r(ra)|^2/r^2*(1 - β^2)^2/(1 - β^2*sin(θ)^2)^3 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Poynting_radiation_per_solid_angle /$ ∂/dΩ(φ) = r^2*|S| = q^2/4/π/c^3*|r(ra)|^2 *(1 - β^2)^2/(1 - β^2*sin(θ)^2)^3 /* Op is angle between r & a /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_16 Poynting_radiation_per_solid_angle /$ ∂/dΩ(φ) = q^2/4/π/c^3*|r(ra)|^2*(1 - β^2)^2/(1 - β^2*sin(θ)^2)^3 /* havent done yet /* havent done yet /********************************************************** /*------> Op is angle between r & a /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_17 Poynting_radiation_per_solid_angle_nonrelativistic /$ ∂/dΩ(φ)(@v< Lucas Power_ Larmor_radiation_total_nonrelativistic /$ P_ Larmor_radiation_nonRel = ∫[dPp,0 to 2*π: ∫[∂(θ), - 1 to 1: q^2/4/π/c^3*a^2*(1 - cos(θ´)^2)]] = 2/3/c^3*q^2*a^2 /* p82h0.5 For the relativistic case of a circular accelerator where v is perpendicular to r such that sinO = 1 or r•v = 0 and a is perpendicular to b = v/c /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_18 Power_ Larmor_radiation_total_nonrelativistic /$ P_ Larmor_radiation_nonRel = 2/3/c^3*q^2*a^2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas /* P_Lienard_circular_accelerator_relativistic /$ ∂/dΩ(P_Lienard) = q^2/4/π/c^3 *a^2*sin(θ´)^2*(1 - β^2)^2/(1 - β^2*sin(θ)^2)^3 = q^2/4/π/c^3*γ^4*a^2*sin(θ´)^2*(1 - β^2) /* where γ = ??????????? /$ P_Lienard = 2/3*q^2 /c^3*γ^4*[a^2 - (βa)^2] /* using the (5-18) integral /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_19 P_Lienard_circular_accelerator_relativistic /$ P_Lienard = 2/3*q^2/c^3*γ^4*[a^2 - (βa)^2] /* havent done yet /* havent done yet /********************* >>> Lucas 6 - Extension of the Universal Force Law to include radiation reaction da/dt /********************************************** ; >>>>>> 6.3 - Derivation of non-relativistic radiation reaction force ; U - electrodynamic potential energy for constant velocity /********************************************************** /*------> Lucas06_01 := Lucas05_04 ; p85h0.3 Note that /$ ∂[∂t: U(r,v) = - v•F(r,v,a) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 06_01 Generalized_potential_U same as Lucas05_04 no need same as Lucas05_04 no need same as Lucas05_04 /********************************************************** /*------> Lucas06_02 := Lucas05_07 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 06_02 Universal_ED_force_with_acceleration same as Lucas05_07 no need - same as Lucas05_07 no need - same as Lucas05_07 /********************************************************** /*------> LucasWork_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic := /$ ∫[∂(t),τ1 to τ2: F_rad•v) = ∫[∂(t),τ1 to τ2: Lucas05_18Larmor_radiation_total_nonrelativistic) = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*a^2) = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*(∂[∂t: v))^2) /* /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 06_03 /$ Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic ∫[∂(t),τ1 to τ2: F_rad•v) = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*(∂[∂t: v))^2) /* havent done yet /* havent done yet /********************************************************** /*------> integrate by parts Lucas06_04 := Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic /$ Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic = ∫[∂(t),τ1 to τ2: F_rad•v) = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*(∂[∂t: v))^2) = 2/3/c^3*q^2* [ - [∂[∂t: v)•v]|τ2 + [∂[∂t: v)•v]|τ1 + ∫[∂(t),τ1 to τ2: d2/dt2(v)•v) ] = 0 + 2/3/c^3*q^2*∫[∂(t),τ1 to τ2: d2/dt2(v)•v) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 06_04 /$ Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic = ∫[∂(t),τ1 to τ2: F_rad•v) = 0 + 2/3/c^3*q^2*∫[∂(t),τ1 to τ2: d2/dt2(v)•v) /* havent done yet /* havent done yet /********************************************************** /*------> Comparing the integrands of the left and right hand sides of the equation, we can identify the experimentally-confirmed non-relativistic radiation reaction force : /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 06_05 F_radiation_reaction_nonrelativistic /$ F_rad = 2/3/c^3*q^2*∂[∂t: a) /* havent done yet /* havent done yet /********************************************** ; >>>>>> 6.4 - Derivation of relativistic radiation reaction force ; /********************************************************** /*------> circular accelerator v is perpendicular to r such that sinO = 1 or r•v = 0 Lucas05_19 := P_Lienard_circular_accelerator_relativistic /$ ∂/dΩ(P_Lienard) = q^2/4/π/c^3 *a^2*sin(θ´)^2*(1 - β^2)^2/(1 - β^2*sin(θ)^2)^3 = q^2/4/π/c^3*γ^4*a^2*sin(θ´)^2*(1 - β^2) /* where γ = ??????????? /$ P_Lienard = 2/3*q^2 /c^3*γ^4*[a^2 - (βa)^2] /* using the (5-18) integral /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 05_19 P_Lienard_circular_accelerator_relativistic /$ P_Lienard = 2/3*q^2 /c^3*γ^4*[a^2 - (βa)^2] /* havent done yet /* havent done yet /********************************************************** /*------> Lucas06_06 := Work_done_on_particle_by_EDF_to_emit_radiation_relativistic /$ ∫[∂(t),τ1 to τ2: F_rad•v) = ∫[∂(t),τ1 to τ2: Lucas05_19: = P_Lienard_circular_accelerator_relativistic) = ∫[∂(t),τ1 to τ2: P_rad) /* from Lucas05_19 /$ = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*γ^4*[a^2 - (βa)^2] /* from Lucas p86h0.75 /$ = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*γ^2*a^2) /* #NOTE!!! Lucas has gamma^2, not gamma^4, dropped 2nd expression!!!! /$ = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*γ^2*∂[∂t: v)•∂[∂t: v)) = 0 + 2/3/c^3*q^2*γ^2*∫[∂(t),τ1 to τ2: d2/dt2(v)•v) /* NOTE : change in integral - I dont trust this!! assuming periodicity in charged particle structure, boundary terms in the integral by parts disappear /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 06_07 F_radiation_circular_accelerator_relativistic /$ F_rad = 2/3/c^3*q^2*γ^2*∂[∂t: a) /* havent done yet /* havent done yet /******************** >>> Lucas 7 - Electrodynamic origin of gravitational forces /********************************************** ; >>>>>> 7.1 - Introduction, Electrodynamic origin of gravitational forces ; /********************************************************** /*------> 07_01 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_01 Commonality_of_electrodynamics_and_gravity /$ F_Coulomb = q1 *q2 /r^2 F_gravity = mg1*mg2/r^2 /* where r = |r2 - r1| /* havent done yet /* havent done yet /********************************************************** /*------> 07_02 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_02 /$ Gμv = 8*π/c*G*Tμv /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Combined_Lorentz_force_Ampere_induction /$ F = q*E0 + v/cBi = q*E0 + v/c(q*v/cE0) = q*E0*(1 + (v/c)^2) /* /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_03 Combined_Lorentz_force_Ampere_induction /$ F_Lorentz_AmpInductn = q*E0*(1 + (v/c)^2) /* havent done yet /* havent done yet /********************************************************** /*------> 07_04 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_04 From_free_electron_drift_in_conductors /$ (v/c)^2 ≈ [ 3*10^(-2)/3*10^8 ]^2 ≈ 10^(-20) /* speeds in m/sec /* havent done yet /* havent done yet /********************************************************** /*------> Lucas07_05 := Lucas05_07 ; Lucas05_07 := Universal_ED_force_with_acceleration /$ F(r,v,a) = = q*q´/r^2* [ + { (1 - β^2)*r + 2*r^2/c^2*a } / [r^2 - {r(rβ)}^2/r^2]^(1/2) - (1 - β^2)*{ (β•r)*r(rβ) + (r•r)*r(ra/c^2) } / [r^2 - {r(rβ)}^2/r^2]^(3/2) ] /* first acceleration term, a, gives rise to Newtons Second Law (F=m*a) second acceleration term, ra, gives rise to absorption/emission of electromagnetic radiation or light For acceleration a=0 and writing out b=v/c ???????? too many "r"???? /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_05 Universal_ED_force_with_acceleration same as Lucas05_07 no need - same as Lucas05_07 no need - same as Lucas05_07 /********************************************************** /*------> ?? conversion of [r^2 - {r(rb)}^2/r^2] to (1 - b^2*sin^2(O))^(3/2) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_06 Universal_force_same_as_covariant_relativistic_Maxwell /$ F(r,v,a) = = + q*q´/r^2*(1 - β^2)*r /(1 - β^2*sin(θ)^2)^(1/2) - q*q´/r^2*(1 - β^2)*(r•v/c)*r(rv/c)/(1 - β^2*sin(θ)^2)^(3/2) /* havent done yet /* havent done yet /********************************************** ; >>>>>> 7.2 - Origin of gravitational forces ; Gaussian system of units, assuming constant velocity a=0, b=v/c, binomial expansion of radial term in (7-6), keep only to order b^4, and substitute sin^2(O) = 1 - cos^2(O) /********************************************************** /*------> Lucas Universal ED force, no acceleration a, expanded radial term /$ F(r,v) = + q*q´*r/r^2 *(1 - β^2)*[1 + 1/2*β^2*sin(θ)^2 + (1/2)*(3/2)/2*β^4*sin(θ)^4 + ...] - q*q´ /r^2*(r•β)*r(rβ)*(1 - β^2)*[1 + 3/2*β^2*sin(θ)^2 + (3/2)*(5/2)/2*β^4*sin(θ)^4 + ...] = + q*q´*r/r^2 *[1 - 1/2*β^2 - 1/2*β^2*cos(θ)^2 - 1/8*β^4 - 1/4*β^4*cos(θ)^2 + 3/8*β^4*cos(θ)^4 + ...] /* /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_07 Universal ED force, no acceleration a, expanded radial term /$ F(r,v) = + q*q´*r/r^2 *[1 - 1/2*β^2 - 1/2*β^2*cos(θ)^2 - 1/8*β^4 - 1/4*β^4*cos(θ)^2 + 3/8*β^4*cos(θ)^4 + ...] - q*q´ /r^2*(r•β)*r(rβ)*[1 + 1/2*β^2 - 3/2*β^2*cos(θ)^2 - 3/8*β^4 - 9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4 + ...] /* havent done yet /* havent done yet /********************************************************** /*------> 07_08 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_08 Total force between two neutral dipoles labels : + positive (proton), - negative (electron), dipole1, dipole2 /$ F = F(2 + ,1 + ) + F(2 + ,1 - ) + F(2 - ,1 + ) + F(2 - ,1 - ) /* havent done yet /* havent done yet /********************************************************** /*------> Lucas07_Fig7_2 looks wrong? check later Lucas07_09 is written incorrectly!?!?!? P, P1, P2, O, t1, t2 are undefined!! It seems to me that each interaction should have been specified for the integration! /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_Fig7_2 Neutral dipole geometry and frequencies, Lucas Figure 7-2 /$ r(1 + ,2 + ) = r(2 + ) - r(1 + ), w1 = 2*π*f1, w2 = 2*π*f2 r(1 - ,2 + ) = r(2 + ) - r(1 - ) - A1*cos(w1*t + P1), A1*f1 = v1 r(1 + ,2 - ) = r(1 + ) - r(2 - ) - A2*cos(w2*t + P2), A2*f2 = v2 r(2 - ,1 - ) = r(2 - ) - r(1 - ) - A2*cos(w2*t + P2) - A1*cos(w1*t + P1) /* havent done yet /* havent done yet /********************************************************** /*------> 07_09 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_09 Neutral oscillating dipoles - time averaged force /$ F(r,v) = 1/τ1 *∫[dt1,0 to τ1: 1/τ2 *∫[dt2,0 to τ2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: 1/2/π*∫[∂(φ),0 to 2*π: 1/ π*∫[∂(θ),0 to π: sin(θ) F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v) )))))) /* havent done yet /* havent done yet /********************************************************** /*------> # /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_10 Neutral oscillating dipoles - time averaged force Simplified - assume two collections of dipoles, spherical symmetry (!?!?!) /$ F(r,v) = 1/τ1 *∫[dt1,0 to τ1: 1/τ2 *∫[dt2,0 to τ2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: 1/ π*∫[∂(θ),0 to π: sin(θ) F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v) )))))) /* havent done yet /* havent done yet /********************************************** ; >>>>>> 7.3 - Computation of radial force term ; /********************************************************** /*------> p93h0.5 in normal lab experiments that measure gravity, r2 - r1 >> A1, A2, so we will drop A1, A2 terms /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_11 Neutral oscillating dipoles - time averaged radial-only force /$ F(2 - ,1 + ) = q1*q2*r21/|r2 + A1 - r1|^2*ξ F(2 + ,1 + ) = q1*q2*r21/|r2 - r1|^2*ξ F(2 + ,1 - ) = q1*q2*r21/|r2 - A1 - r1|^2*ξ F(2 - ,1 - ) = q1*q2*r21/|r2 - A1 + A2 - r1|^2*ξ where ξ = 1 - 1/2*(b2 - b1)^2 - 1/2*(b2 - b1)^2*cos(θ)^2 - 1/8*(b2 - b1)^4 - 1/4*(b2 - b1)^4*cos(θ)^2 + 3/8*(b2 - b1)^4*cos(θ)^4 + ... /* havent done yet /* havent done yet /********************************************************** /*------> 07_12 ; p93h0.65 for most laboratory measurements of gravity, b2≈b1 & (b2 - b1)≈0 For this case only A11 and A22 terms are left where q1=q2=e is charge of proton, -e is charge of electron ?????? CHECK THIS!! /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_12 Neutral oscillating dipoles - radial-only force, time averaged, simplified /$ F(q2,q1) = q1*q2*r21/|r2 - r1|^2*ξ /* where /$ ξ = 1 - 1/2*(b2 - b1)^2 - 1/2*(b2 - b1)^2*cos(θ)^2 - 1/8*(b2 - b1)^4 - 1/4*(b2 - b1)^4*cos(θ)^2 + 3/8*(b2 - b1)^4*cos(θ)^4 + ... /* havent done yet /* havent done yet /********************************************************** /*------> p94h0.5 sum of "1 terms" should equal 0 -> wrong sign for one of the "1"s? /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_13 Neutral oscillating dipoles - radial-only force, time averaged, double-simplified /$ F(2 + ,1 + ) = e^2*r21/|r2 - r1|^2*[1] F(2 + ,1 - ) = -e^2*r21/|r2 - r1|^2*[1 - bpe^2*K1 + bpe^4*k2 ] F(2 - ,1 + ) = -e^2*r21/|r2 - r1|^2*[1 - bep^2*K1 + bep^4*k2 ] F(2 - ,1 - ) = -e^2*r21/|r2 - r1|^2*[1 - bee^2*K1 + bee^4*k2 ] /* where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 /* havent done yet /* havent done yet /********************************************************** /*------> 07_14 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_14 Total force between two neutral dipoles labels : + positive (proton), - negative (electron), dipole1, dipole2 /$ F = F(2 + ,1 + ) + F(2 + ,1 - ) + F(2 - ,1 + ) + F(2 - ,1 - ) = -e^2*r21/|r2 - r1|^2 *[+ 2*Bpe *Bep *K1 + 4*Bpe^3*c^2*Bep *k2 + 4*Bpe *Bep^3*c^2*k2 - 6*Bpe^2*c *Bep^2*c *k2 ] /* where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Component forces between two neutral dipoles, radial p95h0.0 example - odd powers of sin(wi*ti + Pi) average to zero /$ 1/τ *∫[∂(t),0 to τ: 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ))) = ω/2/π *∫[∂(t),0 to 2*π/ω: 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ))) /* set x = w*t /$ = 1/2/π *∫[∂(t),0 to 2*π: 1/2/π*∫[∂(φ),0 to 2*π: sin(x + φ))) = 1/2/π *∫[∂(t),0 to 2*π: 1/2/π*∫[∂(φ),0 to 2*π: sin(x)cos(φ) - cos(x)*sin(φ))) = (1/2/π)^2*∫[∂(t),0 to 2*π: ∫[∂(φ),0 to 2*π: sin(x)cos(φ) - cos(x)*sin(φ))) = (1/2/π)^2* ∫[∂(φ),0 to 2*π: sin(x)cos(φ) - cos(x)*sin(φ)) ) /* this latter integral |(0 to 2*π) /$ = (1/2/π)^2*(-2)*(0 - 0) /* 30Aug2015 Howell - I dont like this last step or two, but seems reasonable as result. However, not just equal-powered sin*cos expressions! /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_15 Component forces between two neutral dipoles, radial /$ 1/τ*∫[∂(t),0 to τ: 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ))) = 0 /* havent done yet 30Aug2015 Howell - I dont like this last step or two, but seems reasonable as result. However, not just equal-powered sin*cos expressions! /********************************************************** /*------> Lucas From (7-10) and (7-14) /$ F_G(r,v) = 1/τ1 *∫[dt1,0 to τ1: 1/τ2 *∫[dt2,0 to τ2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: 1/ π*∫[∂(θ),0 to π: sin(θ) F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v) ))))) = w1/2/π*∫[dt1,0 to 2*π/w1: w2/2/π*∫[dt2,0 to 2*π/w2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: 1/ π*∫[∂(θ),0 to π: sin(θ) -e^2*r21/|r2 - r1|^2*6*Bpe^2*Bep^2*k2 ))))) = w1/2/π*∫[dt1,0 to 2*π/w1: w2/2/π*∫[dt2,0 to 2*π/w2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: -e^2*r21/|r2 - r1|^2*6*Bpe^2*Bep^2 *4/15/π )))) = w1/2/π*∫[dt1,0 to 2*π/w1: w2/2/π*∫[dt2,0 to 2*π/w2: -e^2*r21/|r2 - r1|^2*6*Bpe^2*(1/2)^2*4/15/π )))) = -e^2*r21/|r2 - r1|^2*(A1*w1/c)^2*(A2*w2/c)^2*2/5/π = - 2/5/π *e^2*r21/|r2 - r1|^2*(A1*w1/c)^2*(A2*w2/c)^2 /* Note : this is an attractive force ONLY! where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 /* in (7-16) above /$ -sinOdO = -∂(cos(θ)), x=ω*t, dx=ω*∂(t) /* such that /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_16 Force of gravity from neutral dipoles Note : this is an attractive force ONLY! /$ F_G(r,v) = - 2/5/π *e^2*r21/|r2 - r1|^2*(A1*w1/c)^2*(A2*w2/c)^2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas This looks wrong - improper integral /$ 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 1/2/π*∫[∂(φ),0 to 2*π: + sin^2(ω*t)*cos^2(φ) + cos^2(ω*t)*sin^2(φ) + 2*cos(ω*t)*sin(ω*t)*sinP*cosP )) = 1/2/π*(π*cos^2(ω*t) + 0 + π*sin^2(ω*t)) = 1/2 *( cos^2(ω*t) + sin^2(ω*t)) = 1/2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_17 /$ ??? F_G dipoles - ∫sin(ω*t + φ) 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 0 /* havent done yet This looks wrong - improper integral /********************************************************** /*------> Lucas /$ 1/ π*∫[∂(θ),0 to π: sin(θ)*k2) = 1/ π*∫[∂(θ),0 to π: sin(θ)*( - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4)) = - 1/ π*∫[∂(cos(θ)), - 1 to 1: ( - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4)) = - 1/ π*|( cos(θ) , -1 to 1 : (- cos(θ)/8 - 1/4/3*cos^3(O) + 3/8/5*cos^5(O))) = - 1/ π*(- 2/8 - 1/4/*2/3 + 3/8/*2/5) = 4/15/π /* where k2 = - 1/8 - 1/4*cos^2(O) + 3/8*cos^4(O) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_18 /$ F_G dipoles - ∫sinO*k2 1/ π*∫[∂(θ),0 to π: sin(θ)*k2) = 4/15/π /* where /$ k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 /* havent done yet /* havent done yet /********************************************************** /*------> /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_19 Newtons Universal Law of Gravitation (see 7-1) /$ F_G(r) = -G*mg1*mg2*r21/|r2 - r1| F_G(r) = -G*mg1*mg2 /|r2 - r1|^2 /* ERROR! should be as I have it This is WRONG! - missing terms! -> eq*q´ etcor e /********************************************************** /*------> 31Aug2015 Howell /$ G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_20 Gravity - equality of?: Newton versus Universal force & neutral dipoles /$ G*mg1*mg2 = 2/5/π*(A1*w1/c)^2*(A2*w2/c)^2 G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2 /* WRONG!! /********************************************************** /*------> Lucas := For two bodies with N1 and N2 atoms of atomic number Z1 & Z2 this becomes : /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_21 F_G for two bodies with N1 and N2 atoms of atomic number Z1 & Z2 /$ G*mg1*mg2 = 2/5/π*N1*Z1*(A1*w1/c)^2*N2*Z2*(A2*w2/c)^2 /* check later again I DONT LIKE THIS : He has simply stuck in more symbols This means the original form was incomplete or wrong!!! /********************************************** ; >>>>>> 7.4 - Corroborating evidence for radiative decay of gravity ; /********************************************************** /*------> for simplicity assume N1=N2=1, q1=q2=e, w1=w2=w, m1=m2=m of hydrogen, A1=A2=A ≤ size of hydrogen atom /* from wave equation L*f=c, we have w=2*π*f=2*π*c/L /$ G*m^2 ≥ 2/5/π*e^2*(A*ω/c)^4 = 2/5/π*e^2*(A/c)^4*(2*π*c/L)^4 = 2/5/π*e^2 *(2*π*A/L)^4 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_22 Hydrogen dipole Gm^2 term of gravity /$ G*m^2 ≥ 2/5/π*e^2*(2*π*A/L)^4 /* check later again I DONT LIKE THIS : He has simply stuck in more symbols This could mean the original form was incomplete or wrong!!! /********************************************************** /*------> solve for L /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_23 Lambda (L) - wavelength of H2 dipole radiation (gravity) /$ L^4 ≤ 2/5/π*e^2*16*(π*A)^4/(G*m^2) /* check later again Looks OK - going from (7-22) /********************************************************** /*------> 07_24 ; From CRC Handbook of Chemistry & Physics: ; G := 6.67390e-8 ; % cm^3/g/s^2 ; A_max := 0.37e-8 ; % cm ; e := 4.803e-11 ; % g^0.5*cm^1.5/s OR 4.803e-11 statC ; m_e := 1.6726e-24 ; % g ; solve for L /$ L^4 ≤ 2/5/π*e^2*(2*π*A)^4/G/m_e^2 /* From CRC Handbook of Chemistry & Physics: see constants above /$ L^4 ≤ 2/5/π*(4.803e-11 g^0.5*cm^1.5/s)^2 *(2*π*0.37e-8 cm)^4 / 6.67390e-8 cm^3/g/s^2 / (1.6726e-24 ; % g)^2 ≤ 1.46 cm or 14.6 mm /* Howell check /$ lamda := power (2/5/π*(power e 2)*(power (2*π*A_max) 4)/G/(power m_e 2)) 0.5 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_24 lamda /$ L^4 ≤ 1.46 cm or 14.6 mm /* link L^4 ≤ (string power 4 lamda) cm ???OK - same number /********************************************** ; >>>>>> 7.6 - Computation of non-radial gravitational force term ; /********************************************************** /*------> 07_25 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn 07_25 "no_issue ( Neutral oscillating dipoles - non-radial force, ??NOT time averaged, double-simplified Adapt Lucas07_13 for radial-only force, drop the r21 terms ) /$ F(2 + ,1 + ) = e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1] F(2 + ,1 - ) = -e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1 - bpe^2*K1 + bpe^4*k2 ] F(2 - ,1 + ) = -e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1 - bep^2*K1 + bep^4*k2 ] F(2 - ,1 - ) = -e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1 - bee^2*K1 + bee^4*k2 ] /* where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Neutral oscillating dipoles - non-radial force, time averaged, double-simplified Adapt Lucas07_16 /$ F(r,v) = 1/τ1 *∫[dt1,0 to τ1: 1/τ2 *∫[dt2,0 to τ2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: 1/ π*∫[∂(θ),0 to π: sin(θ) F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v) ))))) = w1/2/π*∫[dt1,0 to 2*π/w1: w2/2/π*∫[dt2,0 to 2*π/w2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: 1/ π*∫[∂(θ),0 to π: sin(θ) e^2*(r•β)*r(rβ)/|r2 - r1|^2*6*Bpe^2*Bep^2*k2 ))))) = w1/2/π*∫[dt1,0 to 2*π/w1: w2/2/π*∫[dt2,0 to 2*π/w2: 1/2/π*∫[dP1,0 to 2*π: 1/2/π*∫[dP2,0 to 2*π: e^2*(r•β)*r(rβ)/|r2 - r1|^2*6*Bpe^2*Bep^2*(-3/2/π) )))) = w1/2/π*∫[dt1,0 to 2*π/w1: w2/2/π*∫[dt2,0 to 2*π/w2: e^2*(r•β)*r(rβ)/|r2 - r1|^2 *6*(A1*w1/c)^2*(1/2)*(A2*w2/c)^2*(1/2) *(-3/2/π) )) = - e^2*(r•β)*r(rβ)/|r2 - r1|^2 *(A1*w1/c)^2*(A2*w2/c)^2 *(9/4/π) /* ??????Note : is this an attractive force ONLY??? (see 7-16) where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 /* in (7-12) above (like (7-16) previously) /$ -sinOdO = -∂(cos(θ)), x=ω*t, dx=ω*∂(t) such that /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_26 Neutral oscillating dipoles - non-radial force, time averaged, double-simplified ??????Note : is this an attractive force ONLY??? (see 7-16) /$ F(r,v) = - e^2*(r•β)*r(rβ)/|r2 - r1|^2(A1*w1/c)^2*(A2*w2/c)^2*(9/4/π) /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Adapt Lucas07_18 which is similar /$ 1/ π*∫[∂(θ),0 to π: sin(θ)*k3) = 1/ π*∫[∂(θ),0 to π: sin(θ)*( - 3/8 - 9/4*cos(θ)^2 + 15/8*cos(θ)^4)) = - 1/ π*∫[∂(cos(θ)), - 1 to 1: ( - 3/8 - 9/4*cos(θ)^2 + 15/8*cos(θ)^4)) = - 1/ π*|( cos(θ) , -1 to 1 : (- 3/8*cos(θ) - 9/4*2/3*cos^3(O) + 15/8*2/5*cos^5(O))) = - 1/ π*(- 3/8*2 - 9/4/*2/3 + 15/8/*2/5) = -3/2/π /* where /$ k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 k3 = - 3/8 - 9/4*cos(θ)^2 + 15/8*cos(θ)^4 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn 07_27 "no_issue Determination of ∫sinO*k3 /$ 1/π*∫[∂(θ),0 to π: sin(θ)*k3) = -3/2/π /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Gravity as neutral oscillating dipoles with Universal force remember - this has been double-simplified /$ F_G_total(r,v) = F_G_radial(r,v) + F_G_nonradial(r,v) /* from (7-16) /$ F_G_radial(r,v) = -e^2*r21/|r2 - r1|^2*(A1*w1/c)^2*(A2*w2/c)^2*2/5/π /* from (7-26) /$ F_G_nonradial(r,v) = -e^2*(r•β)*r(rβ)/|r2 - r1|^2 *(A1*w1/c)^2*(A2*w2/c)^2 *(9/4/π) )))) /* therefore /$ F_G_total(r,v) = - 2/5/π*e^2*r21 /|r2 - r1|^2*(A1*w1/c)^2*(A2*w2/c)^2 - 9/4/π*e^2*(r•β)*r(rβ)/|r2 - r1|^2*(A1*w1/c)^2*(A2*w2/c)^2 = [- 2/5/π*e^2*r21 - 9/4/π*e^2*(r•β)*r(rβ)] /|r2 - r1|^2*(A1*w1/c)^2*(A2*w2/c)^2 /* take 2/5/π*r21*e^2 out of [] /$ = 2/5/π*r21*e^2*(A1*w1/c)^2*(A2*w2/c)^2 /|r2 - r1|^2 *[- 1 - 45/8/r21*(r•β)*r(rβ)] /* WRONG expression in Lucas (7-28) : /$ F_G_total(r,v) = G*mg1*mg/|r2 - r1|^2*[- r21 - 45/8*(r21•β)*r21(r21β)] /* Correct form by substituting with corrected (7-20) (as per 31Aug2015 Howell) : /$ G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2 F_G_total(r,v) = G*mg1*mg/|r2 - r1|^2*[- 1 - 45/8/r21*(r•β)*r(rβ)] /* Howells expression Lucas07_27 works OK (its probably in other papers by Lucas - check later) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_28 Gravity as neutral oscillating dipoles with Universal force remember - this has been double-simplified /$ F_G_total(r,v) = G*mg1*mg/|r2 - r1|^2*[- r21 - 45/8*(r21 •β)*r21(r21β)] F_G_total(r,v) = G*mg1*mg/|r2 - r1|^2*[- 1 - 45/8/ r21*(r•β)*r (r β)] /* WRONG expression in Lucas (7-28) : Correct form by substituting with corrected (7-20) its probably in other papers by Lucas - check later) /********************************************** ; >>>>>> 7.8 - Origin of Hubbles Law due to gravitational redshifts ; /********************************************************** /*------> 07_29 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 07_29 Gravitational redshift from star of mass M and radius R ( redshift = -GM/R at stellar surface, zero at infinity G = Newtons universal grav constant /$ z = ∆L/L = G*M/c^2/R /* - confirmed experimentally by Puond&Rebka1960 /* havent done yet /* havent done yet /******************** >>> Lucas 8 - Electrodynamic origin of Inertial forces /********************************************** ; >>>>>> 8.1 - Introduction ; /********************************************************** /*------> Lucas := Ratio of gravitational masses of two objects from Newtons universal gravitational force law /$ F_g = G*mg1*mg2/r12^2 Fg1 = mg1*g = G*mg1*mE/Re^2 Fg2 = mg2*g = G*mg2*mE/Re^2 /* therefore Fg1/Fg2 = mg1/mg2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_01 Ratio of gravitational masses of two objects /$ Fg1/Fg2 = mg1/mg2 /* havent done yet /* havent done yet - but looks straightforward /********************************************************** /*------> Lucas := Ratio of inertial masses of two objects /$ Fi1 = mi1*a1 = mi1*g = mi1 Fi2 = mi2*a2 = mi2*g = mi2 /* therefore Fi1/Fi2 = mi1/mi2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_02 Ratio of inertial masses of two objects /$ Fi1/Fi2 = mi1/mi2 /* havent done yet /* havent done yet - but looks straightforward /********************************************************** /*------> 08_03 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_03 Surface of Earth - inertial forces are equal to gravitational forces /$ Fg1/Fg2 = mg1/mg2 = Fi1/Fi2 = mi1/mi2 /* havent done yet /* havent done yet /********************************************************** /*------> 08_04 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_04 Newtons 2nd law in accelerating reference frame /$ sum(F_real) + Fi = mi*a /* with /$ Fi = -mi*ai /* havent done yet /* havent done yet /********************************************************** /*------> 08_05 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_05 Where mi = mg : /$ Fi = -mi*ai = -mg*ai /* havent done yet /* havent done yet /********************************************** ; >>>>>> 8.2 - Derivation of force of inertia from Universal Force Law ; /********************************************************** /*------> Lucas Universal Force Law - first acceleration term as Newtons 2nd law F=ma Start with (5-7) Universal_ED_force_with_acceleration /$ F(r,v,a) = = q*q´/r^2* [ + { (1 - β^2)*r + 2*r^2/c^2*a } /[r^2 - {r(rβ)}^2/r^2]^(1/2) - (1 - β^2)*{ (β•r)*r(rβ) + (r•r)*r(ra/c^2) } /[r^2 - {r(rβ)}^2/r^2]^(3/2) ] /* Select the acceleration terms only: /$ F(r,v,a) = = q*q´/r^2* [ + 2*r^2/c^2*a /[r^2 - {r(rβ)}^2/r^2]^(1/2) - (1 - β^2)*(r•r)*r(ra/c^2) /[r^2 - {r(rβ)}^2/r^2]^(3/2) ] /* from Lucas05_04 [r^2 - {r(rb)}^2/r^2] = (1 - b^2*sin^2(O)) CHECK this!! ????????????????? /$ = + q*q´/r*2*a/c^2 /[1 - β^2*sin(θ)^2]^(1/2) - q*q´/r*(1 - β^2)*{r(ra/c^2)} /[1 - β^2*sin(θ)^2]^(3/2) = + q*q´/r*2*a/c^2 *[1 + 1/2*β^2*sin(θ)^2] - q*q´/r*(1 - β^2)*{r(ra/c^2)} *[1 - 3/2*β^2*sin(θ)^2 + 3/2*5/2/2*β^4*sin(θ)^4] = + q*q´/r*2*a/c^2 *[1 + 1/2*β^2 - 1/2*β^2*cos(θ)^2] - q*q´/r *{r(ra/c^2)} *[1 + 1/2*β^2 - 3/2*β^2*cos(θ)^2 + 3/8*β^4 - 9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4] /*++++++++++++++++++++++++++++++++++++++ /*add_eqn 08_06 "no_issue Universal Force Law - first acceleration term as Newtons 2nd law F=ma /$ F(r,v,a) = = + q*q´/r*2*a/c^2 *[1 + 1/2*β^2 - 1/2*β^2*cos(θ)^2] - q*q´/r *{r(ra/c^2)} *[1 + 1/2*β^2 - 3/2*β^2*cos(θ)^2 + 3/8*β^4 - 9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4] /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Figure 8-1 is NOT complete in defining the terms! (should define r(2+) etc) however, it is kind of obvious? Example : r(2+) is vector (distance) from reference point to charge q(2+) /$ r(2 + ,1 + ) = r(2 + ) - r(1 + ) and A1*f1 = v1 r(2 + ,1 - ) = r(2 + ) - r(1 - ) - A1*cos(w1*t + ) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn Figure 08-01 "no_issue 08_F8_1 Figure 8-1 Oscillations of electron in vibrating neutral electric dipole /$ ( r(2 + ,1 + ) = r(2 + ) - r(1 + ) and A1*f1 = v1 r(2 + ,1 - ) = r(2 + ) - r(1 - ) - A1*cos(w1*t + ) /* havent done yet /* havent done yet /********************************************************** /*------> /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_07 Universal force time&space-averaged.over.oscillating neutral dipoles Force to be compared with Newtons 2nd law F=ma /$ Fi_neutral_dipoles(r) = 1/τ1 *∫[∂(t),0 to τ1: 1/ π*∫[∂(θ),0 to π: sin(θ) 1/2/π*∫[∂(φ),0 to 2*π: F(r,O,φ,A1,w1,t)]]] /* havent done yet /* havent done yet /********************************************************** /*------> /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_08 Universal force time&space-averaged.over.oscillating neutral dipoles assume spherical symmetry of neutral dipoles, so integral.over.P=2π /$ Fi_neutral_dipoles(r) = 1/τ1 *∫[∂(t),0 to τ1: 1/ π*∫[∂(θ),0 to π: sin(θ) F(r,O,φ,A1,w1,t) ))) /* havent done yet NOTE: key simplification! ignores surface effects locally, but may be important especially for small clusters at outer edge /********************************************** ; >>>>>> 8.3 - Derivation of Newtons 2nd law from 1st acceleration term ; /********************************************************** /*------> Lucas /$ Fi(2 + ,1 + ) = e^2*2*a/|r2 - r1|/c^2*[1] Fi(2 + ,1 - ) = e^2*2*a/|r2 - r1|/c^2*[1 - bpe^2*K1] /* where /$ bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} K1 = (1 + cos(θ)^2)/2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_09 Universal force time&space-averaged.over.oscillating neutral dipoles Force terms from (8-6) for 1st acceleration term to order b^4 /$ Fi(2 + ,1 + ) = e^2*2*a/|r2 - r1|/c^2*[1] Fi(2 + ,1 - ) = e^2*2*a/|r2 - r1|/c^2*[1 - bpe^2*K1] /* where /$ bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} K1 = (1 + cos(θ)^2)/2 /* havent done yet /* havent done yet - Lucas ??NO square of |r2 - r1|? /********************************************************** /*------> Lucas ( Universal force time&space-averaged.over.oscillating neutral dipoles Sum of 1st terms in the [] of the two forces of Lucas08_09 is just 0 /$ F(r,O,φ,A1,w1,t) = F(2 + ,1 + ) + F(2 + ,1 - ) = e^2*2*a/|r2 - r1|/c^2*bpe^2*K1 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_10 Universal force time&space-averaged.over.oscillating neutral dipoles /$ F(r,O,φ,A1,w1,t) = e^2*2*a/|r2 - r1|/c^2*bpe^2*K1 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Universal force time&space-averaged.over.oscillating neutral dipoles Using (8-10) to solve (8-7) ??Lucas said 8-8, but used 8-7, and used P1 not P?? /$ Fi_neutral_dipoles(r) = 1/τ1 *∫[∂(t),0 to τ1: 1/2/π *∫[∂(φ),0 to 2*π: 1/ π *∫[∂(θ),0 to π: sin(θ) F(r,O,φ,A1,w1,t) ))) = 1/τ1 *∫[∂(t),0 to τ1: 1/2/π *∫[∂(φ),0 to 2*π: 1/ π *∫[∂(θ),0 to π: sin(θ) -e^2*2*a/|r2 - r1|/c^2*bpe^2*K1 ))) = 1/τ1 *∫[∂(t),0 to τ1: 1/2/π *∫[∂(φ),0 to 2*π: -e^2*2*a/|r2 - r1|/c^2*bpe^2*(-2/3/π) )) /* substitute /$ τ1 = 2*π/w1 /* to get : /$ = w1/2*π*∫[∂(t),0 to 2*π/w1: -e^2*2*a/|r2 - r1|/c^2*(A1*w1/c)^2*(1/2)*(-2/3/π) ) = 2/3/π*e^2*(A1*w1/c)^2/|r2 - r1|*a = mi1*a /* But here, Lucas does NOT clearly show that /$ mi1 = 2/3/π*e^2*2*(A1*w1/c)^2/|r2 - r1| !! /* This seems more of a STATEMENT of equivalence, but he should show that it is reasonable! /$ (A1*w1/c)^2/|r2 - r1| /* seems more like an acceleration term? From (7-20) : /$ G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2 /* so in that case /$ mg1 ∝ r21^(0.5)*[e*(A1*w1/c)*(A2*w2/c)] /* (I'm not comfortable about the r21 variable) where /$ bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} K1 = (1 + cos(θ)^2)/2 /* ********************* /$ Fi_neutral_dipoles(r) = 2/3/π*e^2*(A1*w1/c)^2/|r2 - r1|*a = mi1*a /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_11 Universal force time&space-averaged.over.oscillating neutral dipoles /$ Fi_neutral_dipoles(r) = 2/3/π*e^2*(A1*w1/c)^2/|r2 - r1|*a = mi1*a /* havent done yet ??Lucas said 8-8, but used 8-7, and used P1 not P?? /********************************************************** /*------> Lucas Used in (8-11) : This is the wrong integral?!? /$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 1/2/π *∫[∂(φ),0 to 2*π: sin^2(ω*t)*cos^2(φ) + 2*cos(ω*t)*sin(ω*t)*sinP*cosP + cos^2(ω*t)*sin^2(φ)) = 1/2/π *(π*cos^2(ω*t) + 0 + π*sin^2(ω*t)) Lucas dropped the integral notation again! = 1/2 *( cos^2(ω*t) + 0 + sin^2(ω*t)) = 1/2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_12 Calculate ∫sin(w*t + P)) /$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 1/2 /* havent done yet Used in (8-11) : This is the wrong integral?!? /********************************************************** /*------> Lucas Machs principle, inertial mass at surface of Earth with respect to the center of the universe Ruc from (8-2) & (7-21) Ratio of inertial masses of two objects /$ Fi1 = mi1*a1 = N1*Z1*(A1*w1/c)^2/Ruc = N1*Z1*(A1*w1)^2/Ruc Fi2 = mi2*a2 = N2*Z2*(A2*w2/c)^2/Ruc = N2*Z2*(A2*w2)^2/Ruc /* therefore /$ Fi1/Fi2 = mi1/mi2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_13 Machs principle, inertial mass at surface of Earth with respect to the center of the universe Ruc /$ Fi1/Fi2 = mi1/mi2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas From derivation of F_gravity (7-16) /$ F_G(r,v) = - 2/5/π*e^2/|r2 - r1|^2*(N1*Z1*A1*w1/c)^2*(N2*Z2*A2*w2/c)^2 = - G*mg1*mg2/|r2 - r1|^2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_14 F_gravity from Universal force time&space-averaged.over.oscillating neutral dipoles /$ F_G(r,v) = - G*mg1*mg2/|r2 - r1|^2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas From Lucas08_01 from Newtons universal gravitational force law /$ F_g = G*mg1*mg2/r12^2 /* Ratio of gravitational masses of two objects /$ Fg1 = mg1*g = G*mg1*mE/Re^2 Fg2 = mg2*g = G*mg2*mE/Re^2 /* therefore /$ Fg1/Fg2 = mg1/mg2 = (N1*Z1*A1*w1)^2*(N2*Z2*A2*w2)^2 = mi1/mi2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_15 Ratio of gravitational masses of two objects /$ Fg1/Fg2 = mg1/mg2 = (N1*Z1*A1*w1)^2*(N2*Z2*A2*w2)^2 = mi1/mi2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas 8-15 shows that gravitational and inertial mass at any point in the universe are equal within a constant k of one another and a radial factor Ruc /$ mg = k*Ruc*mi /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_16 8-15 shows that gravitational and inertial mass at any point in the universe are equal within a constant k of one another and a radial factor Ruc /$ mg = k*Ruc*mi /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Newtons universal gravitational constant G determine from universal force Expression below is over-simplified, and Earths proximity may dominate .over.the average of the spherically symmetric contribution of the rest of the universe /$ G = 9*π*c^4*Ruc^2/10/e^2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_17 Newtons universal gravitational constant G determined from universal force Expression below is over-simplified, and Earths proximity may dominate .over.the average of the spherically symmetric contribution of the rest of the universe /$ G = 9*π*c^4*Ruc^2/10/e^2 /* havent done yet /* havent done yet /********************************************** ; >>>>>> 8.4 - Additions to Newtons 2nd law from 2nd acceleration term ; /********************************************************** /*------> Lucas Additions to Newtons 2nd law from 2nd acceleration term. Lucas07_25 := NOT APPLICABLE? - neutral dipoles, ?non-accelerating? Neutral oscillating dipoles - non-radial force, ??NOT time averaged, double-simplified Adapt Lucas07_13 for radial-only force, drop the r21 terms /$ F(2 + ,1 + ) = e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1] F(2 + ,1 - ) = -e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1 - bpe^2*K1 + bpe^4*k2 ] F(2 - ,1 + ) = -e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1 - bep^2*K1 + bep^4*k2 ] F(2 - ,1 - ) = -e^2*(r•β)*r(rβ)/|r2 - r1|^2*[1 - bee^2*K1 + bee^4*k2 ] /* where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 /* From (8-6) force terms for 2nd a term to order (v/c)^4 given below. for velocity terms in [] expressions, consider b2=b1, leaving only A1w1 terms. Lucas08_06 Universal Force Law - first acceleration term as Newtons 2nd law F=ma. /$ F(r,v,a) = = + q*q´/r*2*a/c^2 *[1 + 1/2*β^2 - 1/2*β^2*cos(θ)^2] - q*q´/r *{r(ra/c^2)} *[1 + 1/2*β^2 - 3/2*β^2*cos(θ)^2 + 3/8*β^4 - 9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4] /* substitute /$ e^2 = q*q´ /* to get : /$ = + e^2/r*2*a/c^2 *[1 + 1/2*β^2 - 1/2*β^2*cos(θ)^2] - e^2/r *{r(ra/c^2)} *[1 + 1/2*β^2 - 3/2*β^2*cos(θ)^2 + 3/8*β^4 - 9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4] /* Screw it - just go with the given formula, look for links to previous equations later. similar to Lucas07_25 but replace r•b with 1, rb with ra/c^2, add 1/c^2 where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k5 = (1 - 3/2*cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 k3 = - 3/8 - 9/4*cos(θ)^2 + 15/8*cos(θ)^4 k6 = + 3/8 - 9/4*cos(θ)^2 - 15/8*cos(θ)^4 /* seems strange - is there a mistake with k6? *********** /$ F(2 + ,1 + ) = e^2*r(ra/c^2)/c^2/|r2 - r1|*[1] F(2 + ,1 - ) = -e^2*r(ra/c^2)/c^2/|r2 - r1|*[1 - bpe^2*k5 - bpe^4*k6 ] /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_18 Additions to Newtons 2nd law from 2nd acceleration term. /$ F(2 + ,1 + ) = e^2*r(ra/c^2)/c^2/|r2 - r1|*[1] F(2 + ,1 - ) = -e^2*r(ra/c^2)/c^2/|r2 - r1|*[1 - bpe^2*k5 - bpe^4*k6 ] /* where /$ bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k5 = (1 - 3/2*cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 k3 = - 3/8 - 9/4*cos(θ)^2 + 15/8*cos(θ)^4 k6 = + 3/8 - 9/4*cos(θ)^2 - 15/8*cos(θ)^4 /* havent done yet Lucas07_25 := NOT APPLICABLE? - neutral dipoles, ?non-accelerating? seems strange - is there a mistake with k6? /********************************************************** /*------> Lucas Additions to Newtons 2nd law from 2nd acceleration term. In (8-18), sum of the first terms in [] of two forces is zero, leaving /$ F = F(2 + ,1 + ) + F(2 + ,1 - ) = e^2*r(ra/c^2)/c^2/|r2 - r1|*[1] -e^2*r(ra/c^2)/c^2/|r2 - r1|*[1 - bpe^2*k5 - bpe^4*k6 ] = -e^2*r(ra/c^2)/c^2/|r2 - r1|*[bpe^2*k5 - bpe^4*k6] /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_19 Additions to Newtons 2nd law from 2nd acceleration term. /$ F = -e^2*r(ra/c^2)/c^2/|r2 - r1|*[bpe^2*k5 - bpe^4*k6] /* havent done yet /* havent done yet /********************************************************** /*------> Lucas For integrals of (8-19), and /$ (1 - 3/2*cos(θ)^2) /* averages to zero. We use terms up to order b^4 for this term. ?Howell - what? Similar to : Lucas08_12 := /$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 1/2/π *∫[∂(φ),0 to 2*π: sin^2(ω*t)*cos^2(φ) + 2*cos(ω*t)*sin(ω*t)*sinP*cosP + cos^2(ω*t)*sin^2(φ)) = 1/2/π *(π*cos^2(ω*t) + 0 + π*sin^2(ω*t)) /* Lucas dropped the integral notation again! /$ = 1/2 *( cos^2(ω*t) + 0 + sin^2(ω*t)) = 1/2 /* For current integral /$ 1 /π *∫[∂(θ),0 to π: sin(O)*((1 - 3*cos(θ)^2/2))) = -1 /π *∫[∂(cos(θ)),0 to π: ((1 - 3*cos(θ)^2/2))) = -1/2/π *(cos - 3/3*cos^3(O))|(@ 0 to π) = -1/2/π *(- 1 - 1 - (-1) + 1) = 0 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_20 Calculation of averaged ∫sin(w*t + P) /$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 0 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Additions to Newtons 2nd law from 2nd acceleration term. /$ F = -e^2*r(ra/c^2)/c^2/|r2 - r1|^2*[bpe^2*k5 - bpe^4*k6] /* As with Lucas08_11 := Using (8-19) in (8-20), instantaneous, non-radial, dipole?? ??Lucas said 8-8, but used 8-7, and used P1 not P?? /$ Fi_neutral_dipoles(r) = 1/τ1 *∫[∂(t),0 to τ1: 1/2/π *∫[∂(φ),0 to 2*π: 1/ π *∫[∂(θ),0 to π: sin(θ)* F(r,O,φ,A1,w1,t) ))) = 1/τ1 *∫[∂(t),0 to τ1: 1/2/π *∫[∂(φ),0 to 2*π: 1/ π *∫[∂(θ),0 to π: sin(θ)* -e^2*r(ra/c^2)/c^2/|r2 - r1|*bpe^4*k6 ))) = 1/τ1 *∫[∂(t),0 to τ1: 1/2/π *∫[∂(φ),0 to 2*π: -e^2*r(ra/c^2)/c^2/|r2 - r1|*bpe^4*3/2/π )) /* substitute /$ τ1 = 2*π/w1 /* to get : /$ = w1/2*π*∫[∂(t),0 to 2*π/w1: -e^2*r(ra/c^2)/c^2/|r2 - r1|*(A1*w1/c)^4*(3/8)*(3/2/π) ) = (w1/2*π)*(2*π/w1)* -e^2*r(ra/c^2)/c^2/|r2 - r1|*(A1*w1/c)^4*(3/8)*(3/2/π) = -9/16/π *e^2*r(ra/c^2)/c^2/|r2 - r1|*(A1*w1/c)^4 = 27/32/π*(A1*w1/c)^4*mi1*r(ra/c^2) where bpp = (b2 - b1) for F(2 + ,1 + ) ≈ 0 bpe = (b2 - b1) for F(2 + ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1)} bep = (b2 - b1) for F(2 - ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)} bee = (b2 - b1) for F(2 - ,1 - ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)} K1 = (1 + cos(θ)^2)/2 k5 = (1 - 3/2*cos(θ)^2)/2 k2 = - 1/8 - 1/4*cos(θ)^2 + 3/8*cos(θ)^4 k3 = - 3/8 - 9/4*cos(θ)^2 + 15/8*cos(θ)^4 k6 = + 3/8 - 9/4*cos(θ)^2 - 15/8*cos(θ)^4 /* seems strange - is there a mistake with k6? *************** /$ Fi_neutral_dipoles_nonNewton2nd = 27/32/π*(A1*w1/c)^4*mi1*r(ra/c^2) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_21 Additions to Newtons 2nd law from 2nd acceleration term. /$ Fi_neutral_dipoles_nonNewton2nd = 27/32/π*(A1*w1/c)^4*mi1*r(ra/c^2) /* havent done yet /* havent done yet /********************************************************** /*------> Lucas /$ 1/ π *∫[∂(θ),0 to π: sin(θ)*k6) = 1/ π *∫[∂(cos(θ)), - 1 to 1: k6) = 1/ π *k6|(from -1 to 1) /* where /$ k6 = + 3/8 - 9/4*cos(θ)^2 - 15/8*cos(θ)^4 = 3/2/π /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_22 Calculation of /$ ∫sinO*k /$ 1/π*∫[∂(θ),0 to π: sin(θ)*k6) = 3/2/π /* havent done yet /* havent done yet /********************************************************** /*------> Lucas /$ 1/2/π*∫[∂(φ),0 to 2*π: sin^4(ω*t + φ)) = 1/2/π*∫[∂(φ),0 to 2*π: [sin(ω*t)*cosP + cos(ω*t)*sinP]^4) = 1/2/π*∫[∂(φ),0 to 2*π: + sin^4(ω*t)*cos^4(φ) + 4*sin^3(ω*t)*cos^3(φ)*cos (ω*t)*sin (φ) + 6*sin^2(ω*t)*cos^2(φ)*cos^2(ω*t)*sin^2(φ) + 4*sin (ω*t)*cos (φ)*cos^3(ω*t)*sin^3(φ) + cos^4(ω*t)*sin^4(φ) ) = 1/2/π*[6*π/8*sin^4(ω*t) +6*π/4*sin^2(ω*t)*cos^2(ω*t) + 6*π/8*cos^4(ω*t)] = 3/8*[sin^2(ω*t) + cos^2(ω*t)]^2 = 3/8 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_23 Calculation of ∫sin^4(w*t + P) /$ 1/2/π*∫[∂(φ),0 to 2*π: sin^4(ω*t + φ)) = 3/8 /* havent done yet /* havent done yet /********************************************************** /*------> /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 08_24 Inertial force law from Universal force /$ F_I = mi*a - 27/32*(A*ω/c)^2*mi*r(ra/c^2) /* havent done yet /* havent done yet /******************** >>> Lucas 9 - Structure and harmony of the universe /********************************************** ; >>>>>> 9.1 - Structure is from symmetry of the Universal Force ; /********************************************************** /*------> Lucas09_01 := Lucas05_04 ; from Lucas05_04 := Generalized_potential_U /$ U(r,v) = q*q´/r*(1 - β^2)/(1 - β^2*sin(θ)^2)^(1/2) = q*q´/r*(1 - β^2)/[r^2 - {r(rβ)/r^2}]^(1/2) /* where /$ β=v/c ∂[∂t: U(r,v) = - v•F(r,v,a) /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 09_01 Generalized_potential_U same as (5-04) no need - same as (5-04) no need - same as (5-04) /********************************************************** /*------> Lucas Lucas05_07 ; from Lucas05_07 := Universal_ED_force_with_acceleration /$ F(r,v,a) = = q*q´/r^2* [ + { (1 - β^2)*r + 2*r^2/c^2*a } / [r^2 - {r(rβ)}^2/r^2]^(1/2) - (1 - β^2)* { (β•r)*r(rβ) + (r•r)*r(ra/c^2) } / [r^2 - {r(rβ)}^2/r^2]^(3/2) ] /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 09_02 Universal_ED_force_with_acceleration same as Lucas05_07 no need - same as Lucas05_07 no need - same as Lucas05_07 /******************** >>> Lucas 10 - Machs principle and the concept of mass /********************************************** ; >>>>>> 10.1 - Inertial mass ; /********************************************************** /*------> Lucas from Lucas08_11 /$ Fi_neutral_dipoles(r) = 2/3/π*e^2*2*(A1*w1/c)^2/|r2 - r1|*a = mi1*a /* therefore /$ mi = 2/3/π*e^2*(A1*w1/c)^2/|r2 - r1| /* But Lucas summarizes this, for a single vibrating neutral dipole, consisting of an atomic electron and nuclear proton, to : /$ mi = 2/3/π*e^2*(A1*w1/c)^2/R/c^2 ???This doesnt make sense??? /* why would ??? : /$ R*c^2 = |r2 - r1| /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_01 Inertial mass of a single dipole of one [proton, electron] /$ mi = 2/3/π*e^2*(A1*w1/c)^2/R/c^2 /* havent done yet ???This doesnt make sense??? /********************************************************** /*------> Lucas for a lump of N atoms each having Z protons and electrons : NOTE : neutrons count too - perhaps he is using neutron = proton+electron if so, should re-emphasize. R is explain but is problematic! /$ mi = N*Z*(2/3/π*e^2/R/c^2)*(A1*w1/c)^2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_02 Inertial mass of a lump of of N atoms of some element with Z protons/ electrons /$ mi = N*Z*(2/3/π*e^2/R/c^2)*(A1*w1/c)^2 /* havent done yet NOTE : neutrons count too - perhaps he is using neutron = proton+electron if so, should re-emphasize. R is explain but is problematic! /********************************************** ; >>>>>> 10.2 - Gravitational mass ; /********************************************************** /*------> Lucas := Factorize radial Fu_G in terms of mi1*mi2*R from (10-1) /$ Fu_G = - (2/5/π)*(e/R)^2*(A1*w1/c)^2*(A2*w2/c)^2*Rh = G*mg1*mg2/R^2*Rh = - 9/10*π*c^4/e^2*Rh *{2/3/π/R*e^2/c^2*(A1*w1/c)^2} *{2/3/π/R*e^2/c^2*(A2*w2/c)^2} = - 9/10*π*c^4/e^2*Rh*mi1*mi2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_03 Factorize radial Fu_G in terms of mi1*mi2*R from (10-1) /$ Fu_G = - 9/10*π*c^4/e^2*Rh*mi1*mi2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas Fug - forces of inertia wrt [center of spiral galaxy mass Mg, center of universe direction R From (10-4) /$ F_I = m*a = m*as*r + m*a0*Rh /* where as = accleration with respect to center of galaxy of mass M in direction r a0 = accleration with respect to center of universe of mass M in direction R ??? does rh -> indicate a unit vector? ??? /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_04 Fug - forces of inertia wrt Rgh direction to center of spiral galaxy mass Mg Ruh direction to center of universe mass Mu /$ F_I = m*a = m*ag*Rgh + m*au*Rh /* where ag or as = accleration with respect to center of galaxy au or a0 = accleration with respect to center of universe /* havent done yet /* havent done yet /********************************************************** /*------> 10_05 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_05 Magnitude of the observed acceleration a when ag << au (= as << a0) /$ a = (ag^2 + au^2)^0.5 = au*(1 + 1/2*ag^2/a0^2 + ...) = au + 1/2*ag^2/a0 + ...) /* havent done yet /* havent done yet /********************************************************** /*------> 10_06 ; /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_06 Force of gravity - wrt center of [universe mass Mu, galaxy mass Mg] /$ F_G = - G*m*Mu*R/R^2 - G*m*Mg*r/r^2 /* havent done yet /* havent done yet /********************************************************** /*------> Lucas For stability, F_Gravity = F_inertia ?? Maybe not - each could balance??? /$ F_I = m*(au + 1/2*ag^2/au + ...) = -F_G = - G*m*Mu*R/R^2 - G*m*Mg*r/r^2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_07 For stability, F_Gravity = F_inertia /$ F_I = m*(au + 1/2*ag^2/au + ...) = -F_G = - G*m*Mu*R/R^2 - G*m*Mg*r/r^2 /* havent done yet ?? Maybe not - each could balance??? /********************************************************** /*------> rough equivalence of first terms in [F_I, F_G] /$ m*(1/2*ag^2/au) = G*m*Mg*r/r^2 /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_08 Approximate, familiar relationship at stability, F_Gravity = F_inertia /$ m*(1/2*ag^2/au) = G*m*Mg*r/r^2 /* havent done yet /* havent done yet /********************************************************** /*------> Solving for acceleration /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_09 acceleration of lump of mass around center of galaxy at stability, F_Gravity = F_inertia /$ as = (2*G*Mg*au)^0.5/r /* havent done yet /* havent done yet /********************************************************** /*------> Use relationship for acceleration in terms of the velocity for circular orbits /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_10 acceleration related to circular orbits at stability, F_Gravity = F_inertia /$ as = vs^2/r = (2*G*Mg*a0)^0.5/r /* havent done yet /* havent done yet /********************************************************** /*------> Solving for vs /*++++++++++++++++++++++++++++++++++++++ /*add_eqn "no_issue 10_11 speed of circular orbits at stability, F_Gravity = F_inertia /$ vs = (2*G*Mg*au)^(1/4) /* havent done yet OK ?OK - constant orbital velocity of planets and moons in a solar system /************************************************* >>> APPENDICES /************************************************* >>>>>> Future extensions of the Universal Force /*$ cat >>"$p_augmented" "$d_Lucas""context/future extensions.txt" /*_endCmd /********************************************** ; >>>>>>>>> Gaussian versus SI units ; /*$ cat >>"$p_augmented" "$d_Lucas""math nomenclature/Gaussian versus SI units.txt" /*_endCmd /********************** >>>>>> Symbol checking and translation - short description /*$ cat >>"$p_augmented" "$d_Lucas""context/symbols [check, translate].txt" /*_endCmd /******************************************** >>>>>>>>> HFLN = Howell's FlatLiner Notation !!!!!!!!!!!!!! 31May2016 /*$ cat >>"$p_augmented" "$d_Lucas""context/Howells flat-line notation short description.txt" /*_endCmd /********************** >>>>>> Document build short description /*$ cat >>"$p_augmented" "$d_Lucas""context/document build short description.txt" /*_endCmd /************************************************* >>>>>> REFERENCES /*$ cat >>"$p_augmented" "$d_Lucas""context/references.txt" /*_endCmd enddoc