Howell - math of Lucas Universal Force, verifications full listing 160613.txt As of : Mon Jun 13 13:59:42 2016 Only the first line of the comments is provided - many equations have long explanations in "Howell - math of Lucas Universal Force.ndf". ******************* Eqn_number : 04_01 Description : Generalized_Amperes_Law Lucas exprn : Bi(r,t) = v/c × E0(Rpcv,t´) Howell exprn : ∇B = 1/c*[4*π*J + dp[dt : E]] Comments : NOTE - I have to check Appendix A later... DIFFERENT from Maxwell equation equivalent! : Doesn't have the curl operation on B, rate of change of E0. HFLN : BIodv(POIo,t) = v/c × E0pcv(POIp,t)) HFLN : ∇BTodv(POIo,t) = 1/c*[4*π*Jodv(POIo,t) + dp[dt : E0pdv(POIp,t)]] ******************* Eqn_number : 04_02 Description : Faradays_Law Lucas exprn : [·dl : E(r´,t´)] = -1/c*dp[dt : [dA : B(r,t) •n]] Howell exprn : [·dl : E(r´,t´)] = - [dA : dp[dt : B(r,t))]·n] Comments : 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!! HFLN : [·dl : Epdv(POIp,t)] = - [dA : dp[dt : BTodv(POIo,t))]·n] ******************* Eqn_number : 04_03 Description : Gauss_Electrostatic_Law Lucas exprn : [da : E(r,t)•nh] = 4*π*Q(particle) Howell exprn : [da : E(r,t)•nh] = 4*π*Q(particle) Comments : OK - same as conventional expression again, which variables have ε0, μ0, π or not, Ill have to check later HFLN : [da : E0pdv(POIp,t)•nh] = 4*π*Q(particle) ******************* Eqn_number : 04_04 Description : Gauss_Magnetostatic_Law Lucas exprn : ▽•B = 0 Howell exprn : ▽•B = 0 Comments : OK - piece of cake... remember - this is for no charge accumulation, steady state electric, magnetic fields!! HFLN : ▽•BTpdv(POIp,t) = ▽•BTpdv(POIo,t) = 0 ******************* Eqn_number : 04_05 Description : Lenz_Induction_Law Lucas exprn : E(r,v,t) ∝ -E0(r,t) = -lambda(v)*E0(r,t) Howell exprn : E(r,v,t) ∝ -E0(r,t) = -lambda(v)*E0(r,t) Comments : Qualitatively OK - but I can't seem to find support for this specific form. HFLN : EIodv(POIp,t) ∝ -E0odv(POIo,t) = -lambda(Vonv)*E0odv(POIo,t) ******************* Eqn_number : 04_05a Description : E as sum of E0 & Ei Lucas exprn : E(r,v,t) = E0(r,t) + Ei(r,v,t) Howell exprn : Didn't look for this yet Comments : I haven't looked for this yet, but Jackson shows "Normal" linear superposition, except solid materials, extreme conditions. HFLN : ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) ******************* Eqn_number : 04_06 Description : Lorentz_Force_Law Lucas exprn : F(r,v,t) = q* E(r,v,t) - v/cB(r,v,t) Howell exprn : F(r,v,t) = q*[ E(r,v,t) + v/cB(r,v,t) ] Comments : WRONG (typo probably)- Lucas is missing the "q" in q*vB(r,v,t)- see (4-06) versus (4-26) and web references HFLN : FLENZodv(POIo,t) = q*[ E(r,v,t) + vB(r,v,t) ] ******************* Eqn_number : 04_07rev1 Description : Galilean_transformation Lucas exprn : Rpcv = ro - vo*t and t´ = t Howell exprn : Rpcv = ro - vo*t and t´ = t Comments : OK - straightforward (eg Jackson 1999 p515h0.55 Eq (11.1) HFLN : Rpcv(POIp) = Rocv(POIo,t) - Vons(particle)*t and t´ = t ******************* Eqn_number : 04_08 Description : Induced_magnetic_flux_density from Amperes law Lucas exprn : Bi(r,v,t) = (v/c)E0(r,t) Howell exprn : Bi(r,v,t) = (v/c)E0(r,t) Comments : OK - Must check derivation of Appendix A Equation (A19) HFLN : BIodv(POIo,t) = (Vons(particle)/c)E0pdv(POIp,t) ******************* Eqn_number : 04_09 Description : Frame_transformation_info_lost by Maxwell Lucas exprn : ∇B(r,v,t) = 4*π/c*J(r,t) + 1/c *∇∫{dr : ∇·J(r,t)/|r - rp|} ≈ 4*π/c*J(r,t) when second term is ignored Howell exprn : ∇B = μ0*J(rp) + μ0/4/π*∇∫{d^3r : ∇·J(r,t)/|r - rp|} ∇B = μ0*J for steady-state magnetic phenomena! Comments : NOT the same!! is this d^3 a typo in Lucas? Lucas versus Jackson - third derivative d^3rp, my typical problem with [c,μ0,ε0,4,π] NOTE : The second expression is for steady-state magnetic phenomena! HFLN : ∇BTodv(POIo,t) = 4*π/c*Jodv(POIo,t) + 1/c *∇∫[∇·Jodv(POIp,t)/|Rpcs(POIo,t) - Rpcs(POIp)|]dr HFLN : ∇BTodv(POIo,t) = μ0*Jpdv(POIp,t) |r - rp| term is intriguing => Vons(particle)*t? ******************* Eqn_number : 04_10 Description : Galilean_transformation Lucas exprn : see Lucas04_07 Howell exprn : see Lucas04_07 Comments : OK - repeat statement, no need to re-check ******************* Eqn_number : 04_11 Description : E&B_fields_static_plus_induced Lucas exprn : E(r,v,t) = E0(r,t) + Ei(r,v,t) B(r,v,t) = B0(r,t) + Bi(r,v,t) Howell exprn : critical distinction between static & induced - have to check later Comments : OK - but where is the effect of the critical distinction between static & induced - have to check later HFLN : ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) HFLN : BTodv(POIo,t) = B0odv(POIo,t) + BIodv(POIo,t), where B0odv(POIo,t) = 0 in Chapter 4 ******************* Eqn_number : 04_12 Description : E Galilean transformation particle to observer frames Lucas exprn : E(rp,tp) = E(ro - vo*t,t) Howell exprn : E(rp,tp) = E(ro - vo*t,t) Comments : OK, EASY - key point, I need to research results and opinions, seems correct HFLN : ETpdv(POIp,t) = ETodv(POIo,t) ******************* Eqn_number : 04_13rev1 Description : Total B magnetic flux density as induced from E0 + Ei Lucas exprn : B(rpv,tp) = Bi(rov - vov*t,t) = (vov/c)[ E0(rpv,tp) + Ei(rpv,tp) ] Howell exprn : B(rpv,tp) = Bi(rov - vov*t,t) = (vov/c)[ E0(rpv,tp) + Ei(rpv,tp) ] Comments : OK simple, but I have questions. 10Jan2016 I assume B0 doesn't appear as there is no "other background" B source - eg permanent magnet of other [charges, currents]. Or is B0 from E0? PROBLEM - When is an induce field "real"? - depends on relative velocities so different observers at different relative velocities see different B fields at the same Point of Interest (POI)!!?? HFLN : BTpdv(POIp,t) = BIodv(POIo,t) = (Vonv(particle)/c)[ E0pdv(POIp,t) + EIpdv(POIp,t) ] ******************* Eqn_number : 04_14 Description : B&E point charge - substituted Amperes law Lucas exprn : B(rp,t) = q/c*(vovrph)/rps^2 + vov Ei(rp,tp) Howell exprn : B(rp,t) = q/c*(vovrph)/rps^2 + vov/cEi(rov,t) Comments : OK - simple, but Lucas is missing c in 2nd term (check Jackson) 10Jan2016 Lucas's units don't balance! (uses Gaussian units...) HFLN : BTpdv(POIp,t) = Q(particle)/c*(Vonv(particle)Rpch(POIo,t))/Rpcs(POIo,t)^2 + Vonv(particle)/cEIpdv(POIp,t) ******************* Eqn_number : 04_14a Description : Point particle and symmetry Lucas exprn : Ei(rpv,tp) = |Ei(rpv,tp)|*rpv/|rpv| Howell exprn : Ei(rpv,t ) = |Ei(rpv,tp)|*rpv/|rpv| Comments : OK, simple, but just use rph = rpv/|rpv|? HFLN : EIpdv(POIp,t) = EIpds(POIp,t)*Rpch(POIo,t) ******************* Eqn_number : 04_15rev3 Description : E,B for symmetry point charge @v_const Adapted non-[derivative, integral] form of Amperes law Lucas exprn : Bi(rov - vov*t,t) = vos/c*Pph * [ q*sinOo*Rocs(POIo)/|rov - vov*t|^3 + sinOo *|Ei(rov - vov*t,t)| ] Howell exprn : Bi(rov - vov*t,t) = vos/c*Pph * [ q*sinOo*Rocs(POIo)/|rov - vov*t|^3 + sinOp *|Ei(rov - vov*t,t)| ] Comments : 09Jan2016 WRONG - I have Op rather than Oo in 2nd term. NOTE : If I repace Op in expression with |Ei(ro - vo*t,t)| Difference between static and induced fields!?! I haven't wrapped my head around the details. HFLN : BIodv(POIo,t) - must re-check basis of angles! = Vons(particle)/c*APpch * [ Q(particle)*sin(AOoc(POIo))*Rocs(POIo)/Rpcs(POIo,t)^3 + sin(AOpc(POIo,t)) *EIods(POIo,t) ] ******************* Eqn_number : 04_16rev4Lucas4-15 Description : E,B for symmetry point charge @v_const Derivative form of Amperes law Lucas exprn : dp[dt : Bi(ro - vo*t,t)] = vos/c*Rocs(POIo)*sinOo*Pph* [ 3*q*vos/c*(Rocs(POIo)*cosOo - vos*t)/|ro - vo*t|^5 + 1/Rocs(POIo)/c*dp[dt : |Ei(ro - vo*t,t)|] ] Howell exprn : dp[dt : BTpdv(POIo,t)] = dp[dt : BTodv(POIo,t)] = Vons(particle)^2/c*sin(AOpc(POIo,t))^2*cos(AOpc(POIo,t))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle) /Rpcs(POIo,t)^3 - EIpds(POIo,t)/Rpcs(POIo,t) - dp[dt : EIpds(POIo,t)]/Vons(particle)/cos(AOpc(POIo,t)) } Comments : 09Jan2016 WRONG! : I use Op not Oo for 2nd term RHS (as with (4-15), different terms) This could partially be due to differentiation, but not all. HFLN : dp[dt : BTpdv(POIo,t)] = dp[dt : BTodv(POIo,t)] = Vons(particle)^2/c*sin(AOpc(POIo,t))^2*cos(AOpc(POIo,t))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle) /Rpcs(POIo,t)^3 - EIpds(POIo,t)/Rpcs(POIo,t) - dp[dt : EIpds(POIo,t)]/Vons(particle)/cos(AOpc(POIo,t)) } It seems to me that Lucas has dropped the "Vons(particle)*cos(AOpc(POIo,t))*EIpds(POIo,t)/Rpcs(POIo,t)" term. But there should be an EIpds(POIo,t) term! ******************* Eqn_number : 04_17rev2 Description : Spherical coordinate transforms Lucas exprn : |r| = |ro - vo*t| = sqrt( (Rocs*sinOo)^2 + (Rocs*cosOo - vos*t)^2 ) vr = v(ro - vo*t) = vr = vos*Rocs*sinOo*Pph Howell exprn : rps = [ (Rocs*sinOo)^2 + (Rocs*cosOo - vos*t)^2 ]^(1/2) vov X rpv = vos*Rocs*sinOo*Pph Comments : 02Jan2016 OK - easy BUT, I still need to check angle basis below... HFLN : Rpcs(POIp) = { [Rocs(POIo,t)*sin(AOoc(POIo,t))]^2 + [Rocs(POIo,t)*cos(AOoc(POIo,t) - Vons(particle)*t]^2 }^(1/2) HFLN : ????Vonv(particle) X Rpcv(POIp) = Vons(particle)*Rocs(POIo,t)*sin(AOoc(POIo))*APph ???? *Rodh(Vonv_X_Rpcv(POIo)) ???? Recheck my HLFN expression! ******************* Eqn_number : 04_18 Description : Faradays law -> B linked by circuit = induced E around circuit Lucas exprn : ∮(·dlp, over C : Eip(rp,tp)) = -1/c*d/dt( ∮(*dap, over S : Bi(ro - vo*t,t)·nh ) Howell exprn : ∮(·dlp, over C : Eip(rp,tp)) = - *d/dt( ∫(*da , over S : Bi(ro - vo*t,t)·nh ) Comments : 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? HFLN : ∮(·dlp, over C : EIpdv(POIp,t)) = -1/c*d/dt( ∮(*da , over S : (BIodv(POIo,t)·nh) ) ******************* Eqn_number : 04_19 Description : E,B for symmetry point charge @v_const add Faradays law to [Ampere] Lucas exprn : ∮(·dlp, over L : (Eip(rp,tp) - v/cBi(rp,tp))) = -1/c* ∮(*dap, over Ap : (dp[dt : Bi(ro - vo*t,t)]·n ) Howell exprn : ∮(·dlp, over L : (Ep (rp,tp) - c/v*Bi(r ,t ))) = -1/c*dp[dt : ∮(*da , over A : B (r,t ]•n ) Comments : WRONG - Notice that I didnt have to apply Stokes theorem! 02Jan2016 My mistake, as this is Stoke's theorem : => Also - better to use ∬s rather than ∮ for Bi over A PROBLEMS - cross-product correction to my deriv, v/cBi(rp,tp versus c/v*Bi(r,t), changed order of diff/integ- B(r,t) vs Bi(ro - vo*t,t), HFLN : Lucas expression ∮(·dlp, over closed curve L : EIpdv(POIp) - v/c*BIodv(POIp,t)) = -1/c*∮(*dap, over Ap : (dp[dt : BIodv(POIo,t)]·n } HFLN : Howell expression ∮(·dlp, over closed curve L : EIpdv(POIp) - c/v*BIodv(POIp,t)) = -1/c *dp[dt : ∮{*da , over A : BIodv(POIo,t)]•n } ******************* Eqn_number : 04_20 Description : Convective_derivative Lucas exprn : d/dt = ∂/∂t + v·∇p Howell exprn : d/dt = ∂/∂t + v·∇p Comments : 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") ******************* Eqn_number : 04_21 Description : convective derivative of Total magnetic flux density Bi Lucas exprn : d/dt[Bi(ro - vo*t,t)] = dp[dt : Bi(ro - vo*t,t)] + ∇[Bi(ro - vo*t,t)v] Howell exprn : d/dt[Bi(ro - vo*t,t)] = dp[dt : Bi(ro - vo*t,t)] + ∇[Bi(ro - vo*t,t)v] Comments : OK - 2nd try gets same as Lucas 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") HFLN : dT[dt : BIodv(POIo,t)] = dp[dt : BIodv(POIo,t)] + ∇[BIodv(POIo,t)Vons(particle)] ******************* Eqn_number : 04_23 Description : Faradays_Law_for_rest_circuit integral form E,B Lucas exprn : ∮(·dlp : Ei (ro - vo*t,t)) = -1/c*∮(*dap : dp[dt : Bi(ro - vo*t,t)]·np) Howell exprn : ∮(·dlp : Eip(ro - vo*t,t)) = -1/c*∮(*dap, over Ap : dp[dt : Bi(ro - vo*t,t)]·n ) Comments : OK - seems good, Note that this is for v=0, but how is this different than (4-2) Faraday's Law? some worry about vector formulae Jackson1999 versus Kreyszig1972 Note : is the prime following n in Lucas04_23 an error? HFLN : ∮(·dlp : EIodv(POIo,t)) = -1/c*∮(*dap, over Ap : dp[dt : BIodv(POIo,t)]·R_A_PI2_odh ) where : Rodh(Vonv_X_Rpcv(POIo)) is the unit normal vector at each point on area A ******************* Eqn_number : 04_24 Description : E&B for [Faradays + part/obs frameTrans] - towards Fu_Faradays_Law Lucas exprn : Ei(rp,vp,tp) = Ei(ro - vo*t,t) + 1/c*[vBi(ro - vo*t,t)] Howell exprn : B(ro - vo*t,t) = (v/c)[ E0(ro - vo*t,t) + Ei(ro - vo*t,t) ] Comments : !!!WRONG - On hold as I haven't been able to "move" (v/c) to the right place.. AND I still must show that Ep = E and Fp = F !! HFLN : Lucas expression EIpdv(POIo,t) = EIodv(POIo,t) + 1/c*[vBIodv(POIo,t)] ******************* Eqn_number : 04_25 Description : Derivation of the Lorentz Force Lucas exprn : Fp(rp,tp) = q*E0(ro - vo*t,t) + q*Ei(ro - vo*t,t) + q/c*[vBi(ro - vo*t,t)] Howell exprn : Fp(rp,tp) = q*E0(ro - vo*t,t) + q*Ei(ro - vo*t,t) + q/c*[vBi(ro - vo*t,t)] Comments : OK - straightforward from (4-24) 12Sep2015 - I still have problems with (4-24) HFLN : FTpdv(POIo,t) = Q(particle)*{ E0odv(POIo,t) + EIodv(POIo,t) + Vons(particle)/cBIodv(POIo,t) } ******************* Eqn_number : 04_26 Description : Derived Lorentz Force F_L(const v : q,E,Bi) Lucas exprn : F (rp,tp) = q*E(ro - vo*t,t) + q/c*[vBi(ro - vo*t,t)] Howell exprn : Fp(rp,tp) = q*E(ro - vo*t,t) + q/c*[vBi(ro - vo*t,t)] Comments : OK - very straightforward HFLN : F_LORENTZpdv(POIo,t) = Q(particle)*{ EIodv(POIo,t) + Vons(particle)/cBIodv(POIo,t) } ******************* Eqn_number : 04_27rev1 Description : Faradays_law_differential_form Lucas exprn : ∇Ei(r - v*t,t) = -1/c*dp[dt : Bi(r - v*t,t)] Howell exprn : ∇′Ei(r - v*t,t) = -1/c*dp[dt : Bi(r - v*t,t)] Comments : OK - Note that this is for v=0 which seems inconsistent with "(r - v*t,t)", and I still unsure that I've properly defined ∇′ Wasn't Stokes theorm already used in Faraday defn for (4-2)(4-18)? If so, might this be a circular argument somehow? (?nothing wrong with that - as it would at least "close the loop" and show consistency). 29May2016 - This is OK => 14Sep2015 p69h0.95 Lucas comment that ET = E0+Ei is obscure to me at present. 13Sep2015 My approach with 1.c was NOT successful (I need to look at it again later) HFLN : ∇′EIodv(POIo,t) = -1/c*dp[dt : BIodv(POIo,t)] ******************* Eqn_number : 04_28arev1 Description : Faradays_law_spherical_coordinates - full form, 1st expression Lucas exprn : ∇Ei(rp,t) = + rh/rs/sinO*[ dp(dO : Ei(rp,t)·Ph*sinO) - dp(dP : Ei(rp,t)·Oh) ] + Oh/rs *[ dp(dP : Ei(rp,t)·rh)/sinO - dp(drs: Ei(rp,t)·Ph*rs) ] + Ph/rs *[ dp(drs : Ei(rp,t)·Oh*rs) - dp(dO : Ei(rp,t)·rh) ] Howell exprn : ∇Ei(ro - vo*t,t) = + rh/rs/sinO *[ dp(dO : Ei(rp,t)·Ph*sinO) - dp(dP : Ei(rp,t)·Oh) ] + Oh/rs *[ dp(dP : Ei(rp,t)·rh)/sinO - dp(drs : Ei(rp,t)·Ph*rs) ] + Ph/rs *[ dp(drs : Ei(rp,t)·Oh*rs) - dp(dO : Ei(rp,t)·rh) ] Comments : OK - straightforward, with a couple of concerns with Lucas's expression - reference frames & notational. Note that I have taken 1/r out of the [] for Oh plus I retain [Oh,Ph] within [] - Lucas simplifies to [Op,Pp] But - shouldn't all angles be primed to get particle/system refFrame? shouldn't there be a hat over last AOpdh in the second term? HFLN : ∇EIodv(POIo,t) = -1/c*dp[dt : BIodv(POIp,t)] = + Rpch(POIo,t) /Rocs(POIo)/sin(AOoc(POIo))*{ dp[dAOoc : EIodv(POIo,t)·ROpdh*sin(AOoc(POIo))] - dp[dAPoc : EIodv(POIo,t) ·ROpdh] } + ROpdh/Rocs(POIo) *{ dp[dAPch : EIodv(POIo,t)·Rpch(POIo,t) /sin(AOoc(POIo))] - dp[dRocs(POIo) : EIodv(POIo,t)*Rocs(POIo)·RPods] } + RPpdh/Rocs(POIo) *{ dp[dRocs(POIo) : EIodv(POIo,t)·ROpdh·ROpdh ] - dp[dAOod : EIodv(POIo,t) ·Rpch(POIo,t) ] } ******************* Eqn_number : 04_28brev1 Description : Faradays_law_spherical_coordinates - full form Lucas exprn : ∇Ei(ro - vo*t,t) = -1/c*dp[dt : B(rp,t)] = -vs/c*rs*sinO*Pp*[ 3*q*vs/c*(rs*cosO - vs*t)/|ro - vo*t|^5 + 1/rs/c*dp[dt : Eis(ro - vo*t,t)] ] Howell exprn : ∇Ei(ro - vo*t,t) -1/c*dp[dt : Bi(ro - vo*t,t)) = -vs/c*rs*sinO*Pp*[ 3*q*vs/c*(rs*cosO - vs*t)/|rs - vs*t|^5 + 1/rs/c*dp[dt : Eis(ro - vo*t,t)] ] Comments : OK - Simple! I used (4-16). 29May2016 - still a concern => But - shouldn't all angles be primed to get particle/system refFrame? HFLN : -1/c*dp[dt : BIodv(POIo,t)) = -Vons(particle)/c*Rocs(POIo)*sin(AOoc(POIo))*RAPpdh *{ 3*Q(particle)*Vons(particle)/c*(Rocs(POIo)*cos(AOoc(POIo)) - Vons(particle)*t)/Rpcs(POIo,t)^5 + 1/Rpcs(POIo,t)/c*dp[dt : EIods(POIo,t)] } ******************* Eqn_number : 04_29arev1 Description : Faradays_law_spherical_coordinates - reduced, integral form Lucas exprn : -1/rs*dp(dO : Ei(ro - vo*t,t))*Pp = -vs/c*rs*sinO*Pp*[ 3*q*vs/c*(r*cosO - v*t)/|ro - vo*t|^5 + 1/rs/c*dp[dt : Eis(ro - vo*t,t)] ] Howell exprn : -1/rs*dp(dO : Ei(ro - vo*t,t))*Pp = -vs/c*rs*sinO*Pp*[ 3*q*vs/c*(r*cosO - v*t)/|ro - vo*t|^5 + 1/rs/c*dp[dt : Eis(ro - vo*t,t)] ] Comments : OK - straightforward, but clouded by some of the text explanations & omissions. HFLN : -1/Rocs(POIo)*dp[dO : EIods(POIo,t)]*RPpdh = -Vons(particle)/c*Rocs(POIo)*sin(AOoc(POIo)) *RPpdh *{ 3*Q(particle)*Vons(particle)/c*(Rocs(POIo)*cos(AOoc(POIo)) - Vons(particle)*t)/Rpcs(POIo,t)^5 + 1/Rocs(POIo)/c*dp[dt : EIods(POIo,t)] } ******************* Eqn_number : 04_29brev1 Description : Faradays_law_spherical_coordinates - dropping term Lucas exprn : 1/r/sinO*∂/∂Pp(Ei(ro - vo*t,t)*Ohp) = 0 Howell exprn : 1/r/sinO*∂/∂Pp(Ei(ro - vo*t,t)*Ohp) = 0 Comments : OK - easy after fixing (4-29a) above HFLN : 1/Rocs(POIo)/sin(AOoc(POIo))*dp[dPp : EIods(POIo,t)*ROpch] = 0 ******************* Eqn_number : 04_30rev4 Description : Lorentz force - Faradays_law_integrated Lucas exprn : Ei(ro - vo*t,t)|Op=Opf - Ei(ro - vo*t,t)|Op=0 = 3*(vs/c*rs)^2*q*∫[dOp, 0 to Opf : (rs*cosOp - vs*t)/|ro - vo*t|^5 *sinOp ] + vs/c*rs ^2 *∫[dOp, 0 to Opf : 1/rs/c*dp[dt : Eis(ro - vo*t,t)]*sinOp ] Howell exprn : Ei(ro - vo*t,t)|Op=Opf - Ei(ro - vo*t,t)|Op=0 = 3*(vs/c*rs)^2*q*∫[dOp, 0 to Opf : (rs*cosOp - vs*t)/|ro - vo*t|^5 *sinOp ] + vs/c*rs ^2 *∫[dOp, 0 to Opf : 1/rs/c*dp[dt : Eis(ro - vo*t,t)]*sinOp ] Comments : OK - simple, but I must redo with "current" derivations I have. Note that the derivative in the second line below has RFo notation (important to keep me on track!). HFLN : EIods(POIo,t=0)|AOtc=AOpc(POIo,t=0) - EIods(POIo,t=0)|AOtc=0 = 3*Q*Vons^2/c^2*Rocs(POIo)^2*∫{dAOtc, 0 to AOpc(POIo,t=0) : (Rocs(POIo)*cos(AOtc(RFp)) - Vons*t)*Rpcs(POIo,t=0)^(-5)*sin(AOtc(RFp )) } + Vons /c *Rocs(POIo)^2*∫{dAOtc, 0 to AOpc(POIo,t=0) : 1/c/Rocs(POIo) *dp[dt : EIods(POIo,t)]*sin(AOpc(POIo,t )) } ******************* Eqn_number : 04_31 Description : F therefore E balance from Lenz's law and symmetry of local forces Lucas exprn : Ei(ro - vo*t,t)|(O=0)*rhp = -L(v)*E0(ro - vo*t,t)*rhp Howell exprn : Ei(ro - vo*t,t)|(O=0)*rhp = -L(v)*E0(ro - vo*t,t)*rhp Comments : OK with concerns - Looks reasonable (see caveats below) and straightforward. No details needed here. Lenz's Law seems to be a very general, only referring to a proportionality between E0 and EI, but 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 HFLN : EIods(POIo,t)|(AOpc=0)*Rpch(POIo,t) = -lambda(v)*E0ods(POIo,t)*Rpch(POIo,t) ******************* Eqn_number : 04_31a Description : Machs principle - Lenz works, SRT & covariant Maxwell fail Lucas exprn : Provided as statement only Howell exprn : not done yet - Important issue for Lucas - I"ll have to think this over... Comments : Machs principle - Important issue for Lucas. not done yet. I"ll have to think this over... ******************* Eqn_number : 04_32rev5 09Jun2016 redo with HFLN & corrected derivatives/integrals, RFt Description : F therefore E balance - simplified (4-30) Lucas exprn : Eis(ro - vo*t,t)|t=0 + L(v)*E0s(ro - vo*t,t)|t=0 = 3 *b*rs *q/|ro - vo*t|^5*{rs/2*sin^2(Op) + vs*t*(cosOp - 1)}|t=0 + b*rs ^2 *∫(dOp, 0 to Opf : 1/rs/c*dp[dt : Eis(ro - vo*t,t)]*sinOp)|t=0 Howell exprn : Eis(ro - vo*t,t)|t=0 + L(v)*E0s(ro - vo*t,t)|t=0 = 3*(b*rs)^2*q/|ro - vo*t|^5*{rs/2*sin^2(Op) + vs*t *sinOp }|t=0 + b*rs ^2 *∫(dOp, 0 to Opf : 1/rs/c*dp[dt : Eis(ro - vo*t,t)]*sinOp)|t=0 Comments : WRONG - (4-23) is for v=0! I have (b*rs)^2 rather than b*rs, and sinO instead of (cosO - 1) Actually - in Lucas (4-33) the r^2 seems to "reappear" !?!? In (4-33), I will use a blend of the [Lucas, Howell] results above. HFLN : EIods(POIo,t=0)|AOtc=AOpc(POIo,t=0) - EIods(POIo,t=0)|AOtc=0 = 3*Q*beta^2*Rocs(POIo)^2*Rpcs(POIo,t=0)^(-5)*{ 1/2*Rocs(POIo)*sin(AOpc(POIo,t=0))^2 + Vons*t*cos(AOpc(POIt,t=0)) } + beta^1*Rocs(POIo)^2*∫{dAOtc, 0 to AOpc(POIo,t=0) : 1/c/Rocs(POIo)*dp[dt : EIods(POIo,t)]*sin(AOpc(POIo,t)) } ******************* Eqn_number : 04_33rev1 Description : F therefore E balance - iterations on (4-32) Lucas exprn : Eis(ro - vo*t,t) APPLY |t=0 TO EACH TERM = K1 + b*rs^2*∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : K1]) + b*rs^2*∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : K2]) where : K1 = 3/2*b^2*q*rs^3/|ro - vo*t|^5*sin^2(Op) - L(v)*q*rs/|ro - vo*t|^3 K2 = b *rs^2*∫(dOp, 0 to Of : 1/rs/c*sinOp*dp[dt : Eis(ro - vo*t,t)]) Howell exprn : Eis(ro - vo*t,t) APPLY |t=0 TO EACH TERM = K1 + b*rs^2*∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : K1]) + b*rs^2*∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : K2]) where : K1 = 3/2*b^2*q*rs^3/|ro - vo*t|^5*sin^2(Op) - L(v)*q*rs/|ro - vo*t|^3 K2 = b *rs^2*∫(dOp, 0 to Opf : 1/rs/c*sinO*dp[dt : Eis(ro - vo*t,t)]) Comments : OK - works great by using a blend of Lucas & Howell expressions for (4-32). But how do I justify each term selected for my hybrid (4-32)? (Lucas dropped [r,B], Howell missed cosO - 1 EXPLAIN : Lucas states p71h0.25 that the v*t*(cosO - 1) are dropped, but doesn"t explain why (maybe v=0? butthat doesn't make sense). Furthermore, I don't obtain the expression v*t*(cosO - 1). HFLN : EIods(POIo,t) APPLY |t=0 TO EACH TERM = K1 + beta*Rpcs(POIo,t)^2*∫(dO, 0 to Of : 1/Rocs(POIo)/c*sinO*dp[dt : K1]) + beta*Rpcs(POIo,t)^2*∫(dO, 0 to Of : 1/Rocs(POIo)/c*sinO*dp[dt : K2]) where : K1 = 3/2*beta^2*Q(particle)*Rocs(POIo)^3/Rpcs(POIo,t)^5*sin^2(AOpc(POIo,t)) - lambda(v)*Q(particle)*Rocs(POIo)/Rpcs(POIo,t)^3 K2 = beta *Rocs(POIo)^2*∫(dO, 0 to Of : 1/Rocs(POIo)/c*sin(AOpc(POIo,t))*dp[dt : EIods(POIo,t)]) ******************* Eqn_number : 04_34rev5 29May2016 Description : F therefore E balance - taking partial derivatives wrt time Lucas exprn : Eis(ro - vo*t,t) APPLY |t=0 TO EACH TERM =+ 3/2*b^2*q*rs^3/|ro - vo*t|^5*sin^2(Op) - L(v)*q*rs/|ro - vo*t|^3 + b *rs^2*∫(dOp, 0 to Opf : sinOp* (+ 15/2*b^3*q*rs^4/|ro - vo*t|^7*sin^2(Op)*cosOp - 3 *b *q*rs^2/|ro - vo*t|^5*L(v) *cosOp )) + b^2 *rs^4*∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : ∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : Eis(ro - vo*t,t)])]) Howell exprn : Eis(ro - vo*t,t) APPLY |t=0 TO EACH TERM =+ 3/2*b^2*q*rs^3/|ro - vo*t|^5*sin^2(Op) - L(v)*q*rs/|ro - vo*t|^3 + b *rs^2*∫(dOp, 0 to Opf : sinOp* (+ 15/2*b^3*q*rs^3/|ro - vo*t|^7*sin^2(Op)*cosOp - 3 *b *q*rs^1/|ro - vo*t|^5*L(v) *cosOp )) + b^2 *rs^4*∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : ∫(dOp, 0 to Opf : 1/rs/c*sinOp*dp[dt : Eis(ro - vo*t,t)])]) Comments : WRONG (Lucas&Howell) - maybe Lucas double-counted an "rs"?, his units DON'T balance. I have r^3r^1 in the middle terms, not r^4&r^2 Lucas dropped vs*t term from (rs*cosO - vs*t) and yet he carries it in the |ro - vo*t| term. He should retain both, or neither. at t=0, rs = rps HFLN : EIods(POIo,t=0)|AOtc=AOpc(POIo,t=0) - EIods(POIo,t=0)|AOtc=0 = + 3/2 *Q*beta^2*Rocs(POIo)^3*Rpcs(POIo,t)^(-5) *sin(AOpc(POIo,t))^2 + beta^1*Rocs(POIo)^2* ∫{dAOtc, 0 to AOpc(POIo,t=0) : sin(AOtc(RFp))^1* ( + 15/2 *Q*beta^3*Rocs(POIo)^3*Rpcs(POIo,t=0)^(-7)*cos(AOtc(RFp))*sin(AOtc(RFp))^2 - 3 *lambda(v)*Q*beta^1*Rocs(POIo)^1*Rpcs(POIo,t=0)^(-5)*cos(AOtc(RFp)) ) } - lambda(v)*Q *Rocs(POIo)^1*Rpcs(POIo,t)^(-3) + beta^2*Rocs(POIo)^4* ∫{dAOtc, 0 to AOpc(POIo,t=0) : 1/c/Rocs(POIo) *sin(AOtc(RFp))^1 dp[dt : ∫{dAOtc, 0 to AOpc(POIo,t=0) : 1/c/Rocs(POIo) *sin(AOtc(RFp))^1* dp[dt : EIods(POIo,t)] ) ] } ******************* Eqn_number : 04_35rev3 Description : F therefore E balance - more iterations Lucas exprn : Ei(r)|t=0 all measures at t=0 =+ E0(rs) *( 3/2*b^2*sin^2(Oo) + 15/8*b^4*sin^4(Oo) ) - E0(rs)*L(v)*(1 + 3/2*b^2*sin^2(Oo)) Howell exprn : Ei(rs,vs) all measures at t=0 =+ E0(rs) *( 3/2*b^2*sin^2(Op) + 15/8*b^4*sin^4(Op) ) - E0(rs)*L(v)*(1 + 3/2*b^2*sin^2(Op)) Comments : OK other than Lucas EI/ET mixup, Oo/Op limit used by Lucas p72h0.0? probably OK, check later E0 is added to get ET in (4-36). As part of the iterative approach, Lucas dropped the last expression with Ei(r). This is in need of [explanation, clarification] - i.e. this is the first-step approximation only. HFLN : EIods(POIo,t=0) = + E0ods(POIo,t=0) *( 3/2*beta^2*sin(AOoc(POIo))^2 + 15/8*beta^4*sin(AOoc(POIo))^4 ) - E0ods(POIo,t-0)*lambda(v)*(1 + 3/2*beta^2*sin(AOoc(POIo))^2) ******************* Eqn_number : 04_36 Description : Er in terms of E0(rs) and L(vs) Lucas exprn : E(rs,vs) = E0(rs) + Ei(rs,vs) all measures at t=0 = E0(rs) *(1 + 3/2*b^2*sin^2(Oo) + 15/8*b^4*sin^4(Oo) ) - E0(rs)*L(vs)*(1 + 3/2*b^2*sin^2(Oo) ) Howell exprn : E(rs,vs) = E0(rs) + Ei(rs,vs) all measures at t=0 = + E0(rs) *(1 3/2*b^2*sin^2(Oo) + 15/8*b^4*sin^4(Oo) ) - E0(rs)*L(vs)*(1 + 3/2*b^2*sin^2(Oo)) Comments : OK - very simple. HFLN : ETodv(POIo,t=0) = E0odv(POIo,t=0) + EIodv(POIo,t=0) = + E0odv(POIo,t=0) *(1 3/2*beta^2*sin^2(AOoc(POIo)) + 15/8*beta^4*sin^4(AOoc(POIo)) ) - E0odv(POIo,t=0)*lambda(v)*(1 + 3/2*beta^2*sin^2(AOoc(POIo))) ******************* Eqn_number : 04_37rev17 10Jun2016 Description : Er_second_iteration Lucas exprn : E(r,v) = E0(r) *(1 + 3/2*b^2*sin^2(O) + 15/8*b^4*sin^4(O) + 35/16*b^6*sin^6(O) + ...) - E0(r)*L(v)*(1 + 3/2*b^2*sin^2(O) + 15/8*b^4*sin^4(O) + 35/16*b^6*sin^6(O) + ...) Howell exprn : ETods(POIo,t=0) = E0ods(POIo,t=0) + EIods(POIo,t=0) = + E0ods(POIo,t=0)* (+ 1 + 3/2*beta^2*sin(AOpc(POIo,t))^2 + 3/4*beta^4*sin(AOpc(POIo,t))^4 + 5/8*beta^6*sin(AOpc(POIo,t))^6 ) - E0ods(POIo,t=0)*lambda(v)* (+ 1 *beta^2*sin(AOpc(POIo,t))^2 + 3/4*beta^4*sin(AOpc(POIo,t))^4 ) Comments : WRONG, but WOW!! This is MUCH closer to Lucas's (4-37), but I can't say that I'm comfortable with my assumptions to get here. REMEMBER : Result ONLY applies to t=0!! Somehow I have an incorrect sequence of factors compared to (1/1,3/2,5/4,7/2,...) that would yield the binomial expansion series! Key assumption that is controversial At t=0, with coincident RFo & RFp : intermediate terms with 1/c and no matching Vons may be dropped. However, I haven't tracked 1/c well, so this could easily be in error. I never did obtain Lucas's (1 - cos(AOpc(RFp))) This needs to be REDONE in the by sticking 1st 2 terms of (4-34) (without EIods(POIo,t=0)) into (4-32) rather than (4-33) as I did previously??!! I want a fundamental expression!! ******************* Eqn_number : 04_38 Description : Binomial_expansion_for_E0_terms Lucas exprn : (1 - b^2*sin^2(O))^(-3/2) = 1 + 3/2*b^2*sin^2(O) + 15/8*b^4*sin^4(O) + 35/16*b^6*sin^6(O) + ... Howell exprn : (1 - b^2*sin^2(O))^(-3/2) = 1 + 3/2*b^2*sin^2(O) - 15/8*b^4*sin^4(O) + 35/16*b^6*sin^6(O) + ... Comments : OK - simple HFLN : (1 - beta^2*sin^2(AOoc(POIo)))^(-3/2) = 1 + 3/2*beta^2*sin^2(AOoc(POIo)) - 15/8*beta^4*sin^4(AOoc(POIo)) + 35/16*beta^6*sin^6(AOoc(POIo)) + ... ******************* Eqn_number : 04_39 Description : E(r,v) for constant velocity, non-point charge, observer reference frame Lucas exprn : E(r,v) = (1 - L(v))*E0(r)/(1 - b^2*sin^2(O))^(3/2) Howell exprn : E(r,v) = (1 - L(v))*E0(r)/(1 - b^2*sin^2(O))^(3/2) Comments : OK - simple HFLN : ETodv(POIo,t) = (1 - lambda(v))*E0odv(POIo,t)/(1 - beta^2*sin^2(AOoc(POIo)))^(3/2) ******************* Eqn_number : 04_40 Description : L(v) expression for Gauss law for electric charge Lucas exprn : 4*π*q = ∬s(da : E(r)nh) Howell exprn : 4*π*q = ∬s(da : E(r)nh) Comments : OK - no need to do asd it can be found from Jackson1999, standard formula HFLN : 4*π*Q(particle) = ∬s(d_area : ETodv(POIo,t)RNpch) ******************* Eqn_number : 04_41 Description : L(v) expression for Gauss law for electric charge Lucas exprn : 4*π*q = ∫( dP, 0 to 2*π : ∫(dO, 0 to π : E0(r)*r*sinO*(1 - L(v))/(1 - b^2*sin^2(O))^(3/2) )) = 4*π*q*(1 - L(v))/(1 - b^2) Howell exprn : havent done yet Comments : PRIORITY PROBLEM - sin,cos terms ******************* Eqn_number : 04_42 Description : Special integral with binomial series (1 - b^2*sin^2(O))^(3/2) Lucas exprn : For Lucas04_41 : as ∫(dO, 0 to 2*π : sinO/(1 - b^2*sin^2(O))^(3/2) ) = 2/(1 - b^2) and E0(r) = q/r^2 Howell exprn : PRIORITY - havent done yet Comments : havent done yet ******************* Eqn_number : 04_43 Description : E&B_fields_self_consistent Lucas exprn : E(r,v) = E0(r)*(1 - b^2)/(1 - b^2*sin^2(O))^(3/2) Bi(r,v) = (v/c)E(r,v) Howell exprn : E(r,v) = E0(r)*(1 - b^2)/(1 - b^2*sin^2(O))^(3/2) B(rp,tp) = (v/c)E(rp,tp) Comments : 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). HFLN : ETodv(POIo,t) = E0odv(POIo,t)*(1 - beta^2)/(1 - beta^2*sin^2(AOpc(POIo,t)))^(3/2) BIodv(POIo,t) = Vons(particle)/cEIodv(POIo,t) ******************* Eqn_number : 04_44rev1 30May2016 Description : F_electromag_total_constant_v Lucas exprn : F(r,v) = q*{E(r,v) + (v/c)Bi(r,v)} = q *E0(r) *(1 - b^2)/(1 - b^2*sin^2(O ))^(1/2) - q*|E0(r)|*(1 - b^2)/(1 - b^2*sin^2(O ))^(3/2)*(b_v·r_h)*r_h(r_hb) Howell exprn : F(r,v) = q* E0(r) *(1 - b^2)/(1 - b^2*sin^2(Op))^(1/2) - q*|E0(r)|*(1 - b^2)/(1 - b^2*sin^2(Op))^(3/2)*(b_v·r_h)*r_h(r_hb_v) where b_v = b*v_h Comments : OK - works, Problem - usage of angle Oo in observer reference frame (RFo) instead of Op in (RFp) Oo = Op really applies ONLY when frames are coincident & aligned - here Lucas should use (RFp) primes (eg Op) should Roch(POIo) below be Rodh(Vonv_X_Rpcv(POIo)) from file "Howell - Background math for Lucas Universal Force, Chapter 4.odt"? HFLN : FTodv(POIo,t) = Q(particle) *E0odv(POIo,t) *(1 - beta^2)/(1 - beta^2*sin^2(AOpc(POIo,t)))^(1/2) - Q(particle) *E0odv(POIo,t) *(1 - beta^2)/(1 - beta^2*sin^2(AOpc(POIo,t)))^(3/2) *(beta_v·Roch(POIo))*Roch(POIo)(Roch(POIo)beta_v)] where beta_v = beta*Vonv(particle) ******************* Eqn_number : 04_45 Description : Vector identities for Lorentz Force derivation Lucas exprn : v/c[v/cE0(r,v)] = (v/c)·E0(r)*(v/c) - (vs/c)^2*E0(r) = (v/c)·E0(r)*[(v·r_h)*r_h/c - r_h(r_hv)/c] - (vs/c)^2*E0(r) Howell exprn : v/c[v/cE0(r,v)] = (v/c)·E0(r)*(v/c) - (vs/c)^2*E0(r) = (v/c)·E0(r)*[(v·r_h)*r_h/c - r_h(r_hv)/c] - (vs/c)^2*E0(r) Comments : 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"? HFLN : Vonv(particle)/c[Vonv(particle)/cE0(r,v)] = (Vonv(particle)/c)·E0odv(POIo,t)*[(Vonv(particle)·Roch(POIo))*Roch(POIo)/c - Roch(POIo)(Roch(POIo)Vonv(particle))/c] - (Vons(particle)/c)^2*E0odv(POIo,t) ******************* Eqn_number : 04_46 Description : Vector_operations used for the Lorentz force Lucas exprn : A(BC) = (A·C)B - (A·B)C v = v - (vr)R + (vr)r = (v·r)r - r(rv) Howell exprn : A(BC) = (A·C)B - (A·B)C v = v - (vr)R + (vr)r = (v·r)r - r(rv) Comments : OK - very simple, from textbooks ******************* Eqn_number : 05_01 Description : ?title? Lucas exprn : same as Lucas04_44 Howell exprn : no need - same as Lucas04_44 Comments : no need - same as Lucas04_44 ******************* Eqn_number : 05_02 Description : Generalized_potential_U Lucas exprn : d/dt(U) = -F(r,v)·v - d/dt(F(r,v))·r Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 05_03 Description : ?title? Lucas exprn : d/dt(U(r,v)) = -F(r,v)·v Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 05_04 Description : Generalized_potential_U Lucas exprn : U(r,v) = q*q/r*(1 - b^2)/[r^2 - {r(rb)/r^2}]^(1/2) where b=v/c and d/dt(U(r,v) = -v·F(r,v,a) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 05_05_pre Description : ?title? Lucas exprn : v·F = q*q/r^2*{ (1 - b^2)*v·r + 2*r^2/c^2*v·a } / [r^2 - {r(rb)}^2/r^2]^(1/2) - q*q *(1 - b^2)* { r(rb)·v/c*(v·r) + r^2*r(ra/c)·b } / [r^2 - {r(rb)}^2/r^2]^(3/2) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 05_07 Description : Universal_ED_force_with_acceleration Lucas exprn : F(r,v,a) = = q*q/r^2* [ + { (1 - b^2)*r + 2*r^2/c^2*a } / [r^2 - {r(rb)}^2/r^2]^(1/2) - (1 - b^2)* { (b·r)*r(rb) + (r·r)*r(ra/c^2) } / [r^2 - {r(rb)}^2/r^2]^(3/2) ] Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 05_08 Description : Phipps_ED_force_relativistic_transverse Wesleys_ED_force_relativistic_circular Lucas exprn : F(r,v,a : r perpendicular to v, a=0) = q*q*r_h/r^2*[(1 - b^2)/(1 - b^2*sin^2(O))^(1/2) ](@O=π/2) ≈ q*q*r_h/r^2* (1 - b^2)^(1/2) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 05_09 Description : Webers_ED_force_relativistic_parallel Lucas exprn : F(r,v,a : v|rs|a) = q*q/rs^3*{ (1 - b^2)*r + 2*r^2/c^2*a + ...} ≈ F_Weber Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 05_10 Description : F_Lienard_Wichert_electric_field_v_muchless_c Lucas exprn : F_Lienard_Wichert v< eq*q etcor e ******************* Eqn_number : 07_20 Description : Gravity - equality of?: Newton versus Universal force & neutral dipoles Lucas exprn : G*mg1*mg2 = 2/5/π*(A1*w1/c)^2*(A2*w2/c)^2 Howell exprn : G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2 Comments : WRONG!! ******************* Eqn_number : 07_21 Description : F_G for two bodies with N1 and N2 atoms of atomic number Z1 & Z2 Lucas exprn : G*mg1*mg2 = 2/5/π*N1*Z1*(A1*w1/c)^2*N2*Z2*(A2*w2/c)^2 Howell exprn : check later again Comments : I DONT LIKE THIS : He has simply stuck in more symbols This means the original form was incomplete or wrong!!! ******************* Eqn_number : 07_22 Description : Hydrogen dipole Gm^2 term of gravity Lucas exprn : G*m^2 ≥ 2/5/π*e^2*(2*π*A/L)^4 Howell exprn : check later again Comments : I DONT LIKE THIS : He has simply stuck in more symbols This could mean the original form was incomplete or wrong!!! ******************* Eqn_number : 07_23 Description : Lambda (L) - wavelength of H2 dipole radiation (gravity) Lucas exprn : L^4 ≤ 2/5/π*e^2*16*(π*A)^4/(G*m^2) Howell exprn : check later again Comments : Looks OK - going from (7-22) ******************* Eqn_number : 07_24 Description : lamda Lucas exprn : L^4 ≤ 1.46 cm or 14.6 mm Howell exprn : L^4 ≤ 8.05213e+12cm Comments : ???OK - same number ******************* Eqn_number : 07_25 Description : Neutral oscillating dipoles - non-radial force, ??NOT time averaged, double-simplified Adapt Lucas07_13 for radial-only force, drop the r21 terms Lucas exprn : F(2+,1+) = e^2*(r·b)*r(rb)/|r2 - r1|^2*[1] F(2+,1-) = -e^2*(r·b)*r(rb)/|r2 - r1|^2*[1 - bpe^2*k1 + bpe^4*k2 ] F(2-,1+) = -e^2*(r·b)*r(rb)/|r2 - r1|^2*[1 - bep^2*k1 + bep^4*k2 ] F(2-,1-) = -e^2*(r·b)*r(rb)/|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^2(O))/2 k2 = - 1/8 - 1/4*cos^2(O) + 3/8*cos^4(O) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 07_26 Description : Neutral oscillating dipoles - non-radial force, time averaged, double-simplified ??????Note : is this an attractive force ONLY??? (see 7-16) Lucas exprn : F(r,v) = - e^2*(r·b)*r(rb)/|r2 - r1|^2(A1*w1/c)^2*(A2*w2/c)^2*(9/4/π) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 07_27 Description : Determination of ∫sinO*k3 Lucas exprn : 1/π*∫(dO, 0 to π : sinO*k3) = -3/2/π Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 07_28 Description : Gravity as neutral oscillating dipoles with Universal force remember - this has been double-simplified Lucas exprn : F_G_total(r,v) = G*mg1*mg/|r2 - r1|^2*[- r21 - 45/8*(r21 ·b)*r21(r21b)] Howell exprn : F_G_total(r,v) = G*mg1*mg/|r2 - r1|^2*[- 1 - 45/8/ r21*(r·b)*r (r b)] Comments : WRONG expression in Lucas (7-28) : Correct form by substituting with corrected (7-20) (its probably in other papers by Lucas - check later) ******************* Eqn_number : 07_29 Description : Gravitational redshift from star of mass M and radius R Lucas exprn : 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 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_01 Description : Ratio of gravitational masses of two objects Lucas exprn : Fg1/Fg2 = mg1/mg2 Howell exprn : havent done yet Comments : havent done yet - but looks straightforward ******************* Eqn_number : 08_02 Description : Ratio of inertial masses of two objects Lucas exprn : Fi1/Fi2 = mi1/mi2 Howell exprn : havent done yet Comments : havent done yet - but looks straightforward ******************* Eqn_number : 08_03 Description : Surface of Earth - inertial forces are equal to gravitational forces Lucas exprn : Fg1/Fg2 = mg1/mg2 = Fi1/Fi2 = mi1/mi2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_04 Description : Newtons 2nd law in accelerating reference frame Lucas exprn : sum(F_real) + Fi = mi*a with Fi = -mi*ai Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_05 Description : Where mi = mg : Lucas exprn : Fi = -mi*ai = -mg*ai Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_06 Description : Universal Force Law - first acceleration term as Newtons 2nd law F=ma Lucas exprn : F(r,v,a) = = + q*q/r*2*a/c^2 *[1 + 1/2*b^2 - 1/2*b^2*cos^2(O)] - q*q/r *{r(ra/c^2)} *[1 + 1/2*b^2 - 3/2*b^2*cos^2(O) + 3/8*b^4 - 9/4*b^4*cos^2(O) + 15/8*b^4*cos^4(O)] Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_F8_1 Description : Figure 8-1 Oscillations of electron in vibrating neutral electric dipole Lucas exprn : r(2+,1+) = r(2+) - r(1+) and A1*f1 = v1 r(2+,1-) = r(2+) - r(1-) - A1*cos(w1*t + ) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_07 Description : Universal force time&space-averaged over oscillating neutral dipoles Force to be compared with Newtons 2nd law F=ma Lucas exprn : Fi_neutral_dipoles(r) = 1/τ1 *∫(dt, 0 to τ1 : 1/ π*∫(dO, 0 to π : sinO 1/2/π*∫(dP, 0 to 2*π : F(r,O,P,A1,w1,t) ))) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_08 Description : Universal force time&space-averaged over oscillating neutral dipoles assume spherical symmetry of neutral dipoles, so integral over P=2π Lucas exprn : Fi_neutral_dipoles(r) = 1/τ1 *∫(dt, 0 to τ1 : 1/ π*∫(dO, 0 to π : sinO F(r,O,P,A1,w1,t) ))) Howell exprn : havent done yet Comments : NOTE: key simplification! ignores surface effects locally, but may be important especially for small clusters at outer edge ******************* Eqn_number : 08_09 Description : Universal force time&space-averaged over oscillating neutral dipoles Force terms from (8-6) for 1st acceleration term to order b^4 Lucas exprn : 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^2(O))/2 Howell exprn : havent done yet Comments : havent done yet - Lucas ??NO square of |r2 - r1|? ******************* Eqn_number : 08_10 Description : Universal force time&space-averaged over oscillating neutral dipoles Lucas exprn : F(r,O,P,A1,w1,t) = e^2*2*a/|r2 - r1|/c^2*bpe^2*k1 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_11 Description : Universal force time&space-averaged over oscillating neutral dipoles Lucas exprn : Fi_neutral_dipoles(r) = 2/3/π*e^2*(A1*w1/c)^2/|r2 - r1|*a = mi1*a Howell exprn : havent done yet Comments : ??Lucas said 8-8, but used 8-7, and used P1 not P?? ******************* Eqn_number : 08_12 Description : Calculate ∫sin(w*t + P)) Lucas exprn : 1/2/π *∫(dP, 0 to 2*π : sin(w*t + P)) = 1/2 Howell exprn : havent done yet Comments : Used in (8-11) : This is the wrong integral?!? ******************* Eqn_number : 08_13 Description : Machs principle, inertial mass at surface of Earth with respect to the center of the universe Ruc Lucas exprn : Fi1/Fi2 = mi1/mi2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_14 Description : F_gravity from Universal force time&space-averaged over oscillating neutral dipoles Lucas exprn : F_G(r,v) = - G*mg1*mg2/|r2 - r1|^2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_15 Description : Ratio of gravitational masses of two objects Lucas exprn : Fg1/Fg2 = mg1/mg2 = (N1*Z1*A1*w1)^2*(N2*Z2*A2*w2)^2 = mi1/mi2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_16 Description : 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 Lucas exprn : mg = k*Ruc*mi Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_17 Description : 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 Lucas exprn : G = 9*π*c^4*Ruc^2/10/e^2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_18 Description : Additions to Newtons 2nd law from 2nd acceleration term. Lucas exprn : 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^2(O))/2 k5 = (1 - 3/2*cos^2(O))/2 k2 = - 1/8 - 1/4*cos^2(O) + 3/8*cos^4(O) k3 = - 3/8 - 9/4*cos^2(O) + 15/8*cos^4(O) k6 = + 3/8 - 9/4*cos^2(O) - 15/8*cos^4(O) Howell exprn : havent done yet Comments : Lucas07_25 := NOT APPLICABLE? - neutral dipoles, ?non-accelerating? seems strange - is there a mistake with k6? ******************* Eqn_number : 08_19 Description : Additions to Newtons 2nd law from 2nd acceleration term. Lucas exprn : F = -e^2*r(ra/c^2)/c^2/|r2 - r1|*[bpe^2*k5 - bpe^4*k6] Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_20 Description : Calculation of averaged ∫sin(w*t + P) Lucas exprn : 1/2/π *∫(dP, 0 to 2*π : sin(w*t + P)) = 0 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_21 Description : Additions to Newtons 2nd law from 2nd acceleration term. Lucas exprn : Fi_neutral_dipoles_nonNewton2nd = 27/32/π*(A1*w1/c)^4*mi1*r(ra/c^2) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_22 Description : Calculation of ∫sinO*k Lucas exprn : 1/π*∫(dO, 0 to π : sinO*k6) = 3/2/π Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_23 Description : Calculation of ∫sin^4(w*t + P) Lucas exprn : 1/2/π*∫(dP, 0 to 2*π: sin^4(w*t + P) ) = 3/8 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 08_24 Description : Inertial force law from Universal force Lucas exprn : F_I = mi*a - 27/32*(A*w/c)^2*mi*r(ra/c^2) Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 09_01 Description : Generalized_potential_U Lucas exprn : same as (5-04) Howell exprn : no need - same as (5-04) Comments : no need - same as (5-04) ******************* Eqn_number : 09_02 Description : Universal_ED_force_with_acceleration Lucas exprn : same as Lucas05_07 Howell exprn : no need - same as Lucas05_07 Comments : no need - same as Lucas05_07 ******************* Eqn_number : 10_01 Description : Inertial mass of a single dipole of one [proton, electron] Lucas exprn : mi = 2/3/π*e^2*(A1*w1/c)^2/R/c^2 Howell exprn : havent done yet Comments : ???This doesnt make sense??? ******************* Eqn_number : 10_02 Description : Inertial mass of a lump of of N atoms of some element with Z protons/ electrons Lucas exprn : mi = N*Z*(2/3/π*e^2/R/c^2)*(A1*w1/c)^2 Howell exprn : havent done yet Comments : NOTE : neutrons count too - perhaps he is using neutron = proton+electron if so, should re-emphasize. R is explain but is problematic! ******************* Eqn_number : 10_03 Description : Factorize radial Fu_G in terms of mi1*mi2*R from (10-1) Lucas exprn : Fu_G = - 9/10*π*c^4/e^2*Rh*mi1*mi2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 10_04 Description : Fug - forces of inertia wrt Rgh direction to center of spiral galaxy mass Mg Ruh direction to center of universe mass Mu Lucas exprn : 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 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 10_06 Description : Force of gravity - wrt center of [universe mass Mu, galaxy mass Mg] Lucas exprn : F_G = - G*m*Mu*R/R^2 - G*m*Mg*r/r^2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 10_07 Description : For stability, F_Gravity = F_inertia Lucas exprn : F_I = m*(au + 1/2*ag^2/au + ...) = -F_G = - G*m*Mu*R/R^2 - G*m*Mg*r/r^2 Howell exprn : havent done yet Comments : ?? Maybe not - each could balance??? ******************* Eqn_number : 10_08 Description : Approximate, familiar relationship at stability, F_Gravity = F_inertia Lucas exprn : m*(1/2*ag^2/au) = G*m*Mg*r/r^2 Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 10_09 Description : acceleration of lump of mass around center of galaxy at stability, F_Gravity = F_inertia Lucas exprn : as = (2*G*Mg*au)^0.5/r Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 10_10 Description : acceleration related to circular orbits at stability, F_Gravity = F_inertia Lucas exprn : as = vs^2/r = (2*G*Mg*a0)^0.5/r Howell exprn : havent done yet Comments : havent done yet ******************* Eqn_number : 10_11 Description : speed of circular orbits at stability, F_Gravity = F_inertia Lucas exprn : vs = (2*G*Mg*au)^(1/4) Howell exprn : havent done yet Comments : ?OK - constant orbital velocity of planets and moons in a solar system