WorkDoc – Background Math.odt Table of Contents ToDos : 1 Dones : 1 Changes required 2 Active changes 2 Discrepancy : Units are wrong!! dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo(t),t)] without use of Lenz's Induction Law 2 On hold changes 2 Discrepancy : Form of expression for "dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)]" 2 Discrepancy : Units are wrong!! dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo(t),t)] without use of Lenz's Induction Law 4 Scalar absolute values, [vector, matrix] norms - simplification of expressions 8 New set 8 Completed 9 OPEN from “/home/bill/Projects/Lucas - Universal Force” : 1. 0_Math symbols.odt 2. Howell - Background math for Lucas Universal Force, Chapter 4.odt 3. Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt 4. WorkDoc - Background Math.odt 5. Howell - Background math, summary listing of Chapter 4 formulae.odt – shows differences between Lucas and my derivations 6. Howell - Figures for Lucas Univerrsal Force.odt (sometimes – not often) ************************************** ToDos : 27Mar2018 Discrepancy : Units are wrong!! dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo(t),t)] without use of Lenz's Induction Law 27Mar2018 - FINALLY found it! : pick up from "I.4 Major discrepancies between my own derivations and those of Lucas " in file "Howell - Background math for Lucas Universal Force, Chapter 4.odt" must first resolve "Discrepancy : Form of expression for "dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)]"" 27Mar2018 - convert ALL OLD symbols in "Howell - math of Lucas Universal Force.ndf" using Howell's Flatliner Notation (HFLN) (Qnial program?) 27Mar2018 - augment "Howell - Background math, summary listing of Chapter 4 formulae.odt" with Lucas chapter 4 equations 27Mar2018 - replace drawings with links in "Howell - Background math for Lucas Universal Force, Chapter 4.odt" still have problems with link function for images! Dones : 22Mar2018 Start with #5 to find where I need to work!?!?? PRE - before appying Lenz's Law or Barnes Iterations LEN - using Lenz's Induction Law, with (1 - lambda(v)) BAR - using Lucas's results from Thomas Barnes iterations ************************************ Changes required I assume that most key issues are in this file, although there are many errors in "Howell - Background math for Lucas Universal Force.odt" (especially notations) should write Q'Nial check program ****************************** Active changes 30Mar2018 (4-16) derivation in spherical coordinates NOTE!!! : use "/media/bill/PROJECTS/Qnial/MY_NDFS/matrix derivatives in [cartesian, cylindrical, spherical] coordinates.ndf" https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates - Table of : Conversion between Cartesian, cylindrical, and spherical coordinates - Table of : Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates - Table of : Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates - Table of : Non-trivial calculation rules div grad ⁡ f ≡ ∇⋅∇f ≡ ∇^2f curl grad ⁡ f ≡ ∇×∇f = 0 div curl ⁡ A ≡ ∇⋅(∇×A) = 0 curl curl ⁡ A ≡ ∇×(∇×A) = ∇(∇⋅A) − ∇^2A ∇^2(f*g) = f*∇^2g + 2*∇f⋅∇g + g*∇^2f 04Apr2018 automated translation : #------> (4-16) 'E,B for symmetry point charge @v_const ' # First try : ?date?., third attempt 06Sep2015, rev1 24Sep2015, # 09Jan2016 rev4Lucas4-15, 22May2016 used my expression from file # "Howell - Background math for Lucas Universal Force, Chapter 4.odt" : 6) 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 clear 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! - has too many "c"'s - has the term "(ros*cosOo - vos*t)/|ro - vo*t|" in the first term in parenthesis. - I have NO real idea of why the (ros*cosOo - vos*t) pops up anyways, and why the cos term all of a sudden (spherical coords does not explain this!) 30Mar2018 do spherical coordinate derivative!?!! NOTE!!! : use "/media/bill/PROJECTS/Qnial/MY_NDFS/matrix derivatives in [cartesian, cylindrical, spherical] coordinates.ndf" add_eqn "Possible_Lucas_error_or_omission '04_16rev4Lucas4-15' ( 'E,B for symmetry point charge @v_const ' 'Derivative form of Amperes law ' ) ( '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)|] ] ' ) ( '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)) ' ' }' ) ( '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!' ) ; ****************************** On hold changes Form of expression for "dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)]" 27Mar2018 - FINALLY found it! : pick up from "I.4 Major discrepancies between my own derivations and those of Lucas " in file "Howell - Background math for Lucas Universal Force, Chapter 4.odt" +-----+ From Lucas's book : 10Jan2017 corrections p68h0.0 Equation (4-16), with my more up-to-date symbols, and removing redundant "c"'s (and using BT = B0 + BI, but B0 = 0) : (4-16) dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)] = Vons(particle)/c*Rpcs(POIo(t))*sin(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle)*Vons(particle)/Rpcs(POIo(t))^5*[Rpcs(POIo(t))*cos(AOpc(POIo(t))) - Vons(particle)*t] - dp[dt : EIpds(POIo(t))]/Rpcs(POIo(t)) } = Vons(particle)/c*Rpcs(POIo(t))*sin(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle)*Vons(particle)/Rpcs(POIo(t))^5*Rpcs(POIo(t))*cos(AOpc(POIo(t))) - 3*Q(particle)*Vons(particle)/Rpcs(POIo(t))^5*Vons(particle)*t - dp[dt : EIpds(POIo(t))]/Rpcs(POIo(t)) } = Vons(particle)/c*sin(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle)*Vons(particle) *cos(AOpc(POIo(t)))/Rpcs(POIo(t))^3 - 3*Q(particle)*Vons(particle)^2*t /Rpcs(POIo(t))^4 - dp[dt : EIpds(POIo(t))] } = Vons(particle)^2/c*sin(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle)*cos(AOpc(POIo(t))) /Rpcs(POIo(t))^3 - 3*Q(particle)*Vons(particle) *t/Rpcs(POIo(t))^4 - dp[dt : EIpds(POIo(t))]/Vons(particle) } = Vons(particle)^2/c*sin(AOpc(POIo(t)))*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle) /Rpcs(POIo(t))^3 - 3*Q(particle) /Rpcs(POIo(t))^3 *Vons(particle) *t/cos(AOpc(POIo(t)))/Rpcs(POIo(t)) - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)] = Vons(particle)^2/c*sin(AOpc(POIo(t)))*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle) /Rpcs(POIo(t))^3 *[1 - Vons(particle)*t/cos(AOpc(POIo(t)))/Rpcs(POIo(t)) ] - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } Summarizing : 1) dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)] = Vons(particle)^2/c*sin(AOpc(POIo(t)))*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle)/Rpcs(POIo(t))^3 *[1 - Vons(particle)*t/cos(AOpc(POIo(t)))/Rpcs(POIo(t)) ] - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } +--+ 15&16May2016 But looking at : 29Mar2018 fixes Vons(particle) *t/cos(AOpc(POIo(t)))/Rpcs(POIo(t)) From "R_O0_pcs(POIo(t))", RFo basis : 1)* R_O0_pcs(POIo(t)) = Rocs(POIo)*cos(AOoc(POIo)) - Vons(particle)*t From "R_O0_pcs(POIo(t))", FRp basis 1)** R_O0_pcs(POIo(t)) = Rpcs(POIo(t))*cos(AOpc(POIo(t))) combining (1)* & (1)** : Rocs(POIo )*cos(AOoc(POIo)) - Vons(particle)*t = Rpcs(POIo(t))*cos(AOpc(POIo(t))) So : Vons(particle)*t = Rocs(POIo)*cos(AOoc(POIo)) - Rpcs(POIo(t))*cos(AOpc(POIo(t))) or : Vons(particle)*t/cos(AOpc(POIo(t)))/Rpcs(POIo(t)) = Rocs(POIo)*cos(AOoc(POIo))/Rpcs(POIo(t))/cos(AOpc(POIo(t))) - 1 2) = RO0ocs(POIo)/RO0pcs(POI,t) - 1 Subbing (2) into (1) : 1) dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)] = Vons(particle)^2/c*sin(AOpc(POIo(t)))*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle) /Rpcs(POIo(t))^3 *[1 - Vons(particle)*t/cos(AOpc(POIo(t)))/Rpcs(POIo(t)) ] - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } = Vons(particle)^2/c*sin(AOpc(POIo(t)))*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle) /Rpcs(POIo(t))^3 *[ 1 - RO0ocs(POIo)/RO0pcs(POI,t) - 1 ] - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } Summarizing : 3) dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)] = Vons(particle)^2/c*sin(AOpc(POIo(t)))*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(particle) /Rpcs(POIo(t))^3 *RO0ocs(POIo)/RO0pcs(POI,t) - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } +-----+ Compare this to : dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo(t),t)] without use of Lenz's Induction Law Howell's expression Section "dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo(t),t)] without use of Lenz's Induction Law (need to RE-CHECK!!!)" 6)* dp[dt : BTpdv(POIo(t),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))) } While the factor of 3 IS EXPLAINED, this is NOT the same as (3) in the sub-sub-section immediately above, as I have in (6)* an extra term for dp[dt : E0] at right angles to Rodh(Vonv_X_Rpcv(POIo)), due to angular changes. However, it's hard to tell given the term EIpds(POIo(t))/Rpcs(POIo(t)). By applying Lenz's Law : From "Lenz's Induction Law and it's context" based on Lucas p70h0.85 Equation (4-31) : (4-31) EIpds(POIo(t))*Rpch(POIo(t)) = -lambda(v)*E0ods(POIo)*Rpch(POIo(t)) and therefore : EIpds(POIo(t)) = -lambda(v)*E0ods(POIo) From "E0odv(POIp(t),t) = E0pdv(POIp,t)" : 1)* E0pdv(POIp) = E0odv(POIp(t),t) = Q(particle)/Rpcs(POIp)^2*Rpch(POIp) Subbing (1)* into (4-31)* : (4-31) EIpds(POIo(t))*Rpch(POIo(t)) = -lambda(v)*E0ods(POIo)*Rpch(POIo(t)) 1) = -lambda(v)*Q(particle)/Rpcs(POIp)^2*Rpch(POIo(t)) Subbing (1) into (6)* : dp[dt : BTpdv(POIo(t),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))) } = 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 - (-lambda(v)*Q(particle)/Rpcs(POIp)^2)/Rpcs(POIo(t)) - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } Do I have a mistake with the sign here? Summarizing to yield Howell's expression Section "dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo(t),t)] subbing for Lenz's Law : 2) dp[dt : BTpdv(POIo(t),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 *[1 + lambda(v)/3] - dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t))) } Substituting for lambda from Lucas p73h0.7 in the text : lambda = beta^2 = (Vons(particle)/c)^2 +-----+ 22Mar2018 Constant "c" (speed of light in a vacuum) should be REMOVED from all expressions!!! But ADD magnetic permittivity and electric permeability Lucas uses ?name? Unit system, so "c" appears, but this only confuses my work and it is a red herring. +-----+ 16Mar2018 Wheree am I? Where am I going? From sub-section "On hold changes" : see section "ETods(POIo,t) = E0 - EI -> automatically takes care of Lenz's Law (opposite directions) check this and downstream... 19Dec2017 OK - I get it now after coming back after ~1 year. This is required for the SCALAR equations! Checking E,B - this is a very long process! for later .... +--+ Wheree am I? Start with section "ETods(POIo,t) = E0 - EI -> automatically takes care of Lenz's Law (opposite directions) +--+ Where am I going? #-----+ see section "ETods(POIo,t) = E0 - EI -> automatically takes care of Lenz's Law (opposite directions) check this and downstream... 19Dec2017 OK - I get it now after coming back after ~1 year. This is required for the SCALAR equations! Checking E,B - this is a very long process! for later .... 16Mar2018 start these corrections Section "ETods(POIo,t) = ETpds(POIo(t),t) " 4) ETods(POIo(t)) = |Q(particle)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(AOoc(POIo))*Vons(particle)*t + [Vons(particle)*t]^2 } - EIods(POIo(t)) Search ETods(POIo(t)) >> This doesn't appear anywhere else!?!??? #-----+ 09Jan2017 CAUTION : For far too long, I ignored the lower integration limit, which must be subtracted for definite integrals, and many errors resulted. In most cases, the lower limit does give zero - for example when sin(theta) terms arise for theta=0. #--+ From "R_O0_pcs(POIo,t)" : RFo basis : 1)* R_O0_pcs(POIo,t) = Rocs(POIo)*cos(AOoc(POIo)) - Vons(particle)*t From "R_O0_pcs(POIo,t)" : FRp basis 1)** R_O0_pcs(POIo,t) = Rpcs(POIo,t)*cos(AOpc(POIo,t)) From "R_O0_pcs(POIo,t)" R_O0_ocs(POIo) 1) R_O0_ocs(POIo) = Rocs(POIo)*cos(Oocs(POI)) R_O0_pcs(POIo,t) (RFp) basis 1) R_O0_pcs(POIo,t) = Rpcs(POIo,t)*cos(AOpc(POIo,t)) (RFo) basis 1) R_O0_pcs(POIo,t) = Rocs(POIo)*cos(AOoc(POIo)) - Vons(particle)*t +-----> 08Jun2016 - Can I prove that the term with Vons << the second term, so that it may be dropped? (I doubt it very much!! except for very special conditions!) # General pattern from derivative derivations below : dp[dt : Rpcs(POIo,t)^(-b)*sin(AOpc(POIo,t))^a ] = b*Vons(particle)*Rpcs(POIo,t)^(-b-1)*cos(AOpc(POIo,t))*sin(AOpc(POIo,t))^(a ) + a *Rpcs(POIo,t)^(-b )*cos(AOpc(POIo,t))*sin(AOpc(POIo,t))^(a-1) <-----+ #-----+ 14Jun2016 Calculus of RFt dp[dAOpc : Rpcs(POIo,t=0) ] = dp[dAOpc : |Rpcv(POIo,t)|] (is this wrong?) NOTE : This should be done by a proper vector derivative approach!! 14Jun2016 - Is this WRONG??? #-------+ Don't induced E & B chage direction every other iteration? endsection Scalar absolute values, [vector, matrix] norms - simplification of expressions I need to fix this!! : From "Scalar absolute values, [vector, matrix] norms - simplification of expressions" : scalar norms - if all terms are scalars - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| <> != sum (|x1|,|x2|,|x3|,...) in general (although it may be true in some cases, eg all +ve) vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| <> != product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| <> != sum (|x1|,|x2|,|x3|,...) in general (although it may be true in some cases, eg all have same direction, +ve) From "Lenz's Induction Law and it's context" : 1)* EIodv(POIo,t) = -lambda(v)*E0odv(POIo,t) where lambda(v) is a positive real function of speed New set +-----+ (RFp) basis +-----+ (RFo) basis Completed 17May2016 equation (4-1) - for Chapter 4 situation - perhaps the curl resolves to one dimension ONLY??? Still would yield derivative of B !!! 19Mar2016 I dismantled "Relating [Rpcs,rO0pcs,ROPI2pcs,sin(Opc),cos(Opc)]@t to [Roc,Oo,Po] for (POIo)", and started to re-number equations. All references through the text must be updated! This must also be done for the (RFp) refence frame equations! 20Mar2016 "III. Chapter 4 - Expressions for a POI fixed in the observer reference frame (RFo)" - Need to check that proper vector derivations were used! 22Mar2016 - I still need to get the signs right for the vector cross product "right hand rule", for charge, and some other concepts. 23Mar2016 - Lenz's Induction Law and it's context : Equations were all renumbered, aumented by new equations. Need to changer references in rest of text. 24Mar2016 IT'S OK ! "Lucas p67h0.9 (4-15) - Ppca(POIo,t) & Poca(POIo) should not be in (4-15)" It's NOT Ppca(POIo,t), it's Rpch(POIo,t) where the angle is Vonv(particle) X Rpch(POIo,t) !!! 25Mar2016 I must go back and REDO DERIVATIVES involving unit vectors. Some may require converting a unit vector to a vector divided by its magnitude (v/|v|), resulting in a long and more detailed derivative derivation. See "dp[dt : ETodv(POIo,t)] = dp[dt : ETpdv(POIo,t)], using Lenz's Induction Law" for an example. 26Mar2016 Imporant note to add to overal math descriptions. For now, I put it in "Howell - Background math for Lucas Universal Force, Chapter 4.odt", sub-section "Superluminal speeds". 29Mar2016 - fix inexact unit vector notations in Lenz basics & derivatives (look at earlier work with RDEodv, OPI2oda, etc etc.) example from "Howell - Background math, summary listing of Chapter 4 formulae.odt" "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo,t)] " : PRE derviation : RDEpdh(POIo,t,dt) 14Jun2016 finally done after several months!! These four derivations are related, and currently in my work a [typo, error] by Lucas in one may end up being reversed in later equations • 04_32rev3 WRONG - I have (b*rs)^2 rather than b*rs, Ei,E0 rather than Eis,E0s • 04_33rev1 OK - works great by using a blend of Lucas & Howell expressions. I should redo with new notations. • 04_34rev3 WRONG - I have r^2&r^0 in the middle terms, not r^4&r^2 'As with (4-33), Lucas''s result may be a typo/missing term, so I should redo all.' '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. ' 'Also, Lucas''s units won''t balance?) ' • 04_37rev10 PRIORITY to address!!! 03Jan2016 30Mar2016 ?Inconsistency? : Lenz's Induction Law versus Barnes iterations - I have been assuming : ET = E0 + EI = E0 - lambda(v)*E0 = (1 - lambda(v))*E0 = (1 - beta^2)*E0 but in the end Lucas uses : ET = E0*(1-beta^2)/(1 - beta^2*sin(theta)^2)^(3/2) This is not consistent!!! see (4-31) to (4-43) 15May2016 Simply due to Thomas Barnes iterations !!! 16May2016 done dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo,t)] without use of Lenz's Induction Law (need to RE-CHECK!!!) Factor of 3 in "3*q*vos/c*(ros*cosOo - vos*t)/|ro - vo*t|^5" (actually expression for dp[dt : BT]) My expression sstill doesn't agree with Lucas - but its "close" (sort of) 24Mar2016 - same point as "Factor of 3" This is NOT the same result for dp[dt : BTpdv(POIo,t)] as Lucas p68h0.0 Equation (4-16) : (4-16) dp(dt : Bi(ro - vo*t,t)) = dp(dt : BT(ro - vo*t,t)) = vos/c*ros*sinOo*Pph *[ 3*q*vons/c*(ros*cosOo - vons*t)/|ro - vons*t|^5 + 1/ros/c*dp(dt : |Ei(ro - vo*t,t)|) ] = vos/c*sinOo*Pph* [ 3*q*vons/c*(ros*cosOo - vos*t)/|ro - vons*t|^4 + 1/c*dp(dt : |Ei(ro - vo*t,t)|) ] It seems clear to me that Lucas has : - dropped the "Vons(particle)*cos(Opca(POIo,t))*EIpds(POIo,t)/Rpcs(POIo,t)" term. But there should be an EIpds(POIo,t) term! - has too many "c"'s - has the term "(ros*cosOo - vos*t)/|ro - vo*t|" in the first term in parenthesis. - I have NO real idea of why the (ros*cosOo - vos*t) pops up anyways, and why the cos term all of a suddden (spherical coords does not explain this!) If I : • replace "(ros*cosOo - vos*t)/|ro - vo*t|" with "1" (for an observer reference frame (RFo) on the line of trajectory of the particle), • remove the extra "c"s, and • use my nomenclature, then Lucas's expression becomes : (4-16Mod) dp(dt : BTpdv(ro - vo*t,t)) = Vons(particle)/c*sin(Opca(POIo,t))*Rodh(Vonv_X_Rpcv(POIo)) *[ 3*Q(particle)*Vons(particle)/Rpcs(POIo,t)^3 + dp[dt : EIpds(POIo,t)] ] = Vons(particle)^2/c*sin(Opca(POIo,t))*Rodh(Vonv_X_Rpcv(POIo)) *[ 3*Q(particle)/Rpcs(POIo,t)^3 + dp[dt : EIpds(POIo,t)]/Vons(particle) ] This does not agree with my expression for "dp[dt : BTpdv(POIo,t)] = dp[dt : BTodv(POIo,t)]". 28May2016 - Instead of changing [nomenclature/, symbols] for ALL equations in the file "", I only added a "Howell FlatLiner Notation" (HLFN) version to the file "Howell - math of Lucas Universal Force.ndf" 04_44 Redone & fixed up 30May2016 NUTS!! Improper, indefinite integrals - most results only have sin terms, in which case the lower result = 0. However, (4-32) has a cos term, so the lower result cannot be ignored!!! #----->>> 15Jun2016 Is my expression for dp[dt : E0pds(POIo,t=0)] incorrect? From "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo,t)] " 6*) dp[dt : E0pdv(POIo,t)] = Q(particle)*Vons(particle)/Rpcs(POIo,t)^3 *[ sin(AOpc(POIo,t))*RDEpdh(POIo,t,dt) + 2*cos(AOpc(POIo,t))*Rpch(POIo,t) ] So if I took ONLY the Rpch(POIo,t) component : dp[dt : E0pdv(POIo,t)] = Q(particle)*Vons(particle)/Rpcs(POIo,t)^3*2*cos(AOpc(POIo,t))*Rpch(POIo,t) = 2*Q(particle)*Vons(particle)/Rpcs(POIo,t)^3*cos(AOpc(POIo,t))*Rpch(POIo,t) and dp[dt : E0_Rpch_pds(POIo,t)] = dp[dt : |E0_Rpch_pdv(POIo,t)] = 2*|Q(particle)|*Vons(particle)/Rpcs(POIo,t)^3*|cos(AOpc(POIo,t))| Compare this to "dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)] - proper E0odv(POIo,t) vector approach" : 13*) dp[dt : E0pds(POIo,t) ] = 2*|Q(particle)|*Vons(particle)*cos(AOpc(POIo,t))/Rpcs(POIo,t)^3 = 2*|Q(particle)|*Vons(particle)/Rpcs(POIo,t)^3*cos(AOpc(POIo,t)) SOMETHING'S WRONG!!! - these cannot be the same! Note : This is the SAME as the Rpch component of " dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo,t)]" : 6*) dp[dt : E0pdv(POIo,t)] = Q(particle)*Vons(particle)/Rpcs(POIo,t)^3 *[ sin(AOpc(POIo,t))*RDEpdh(POIo,t,dt) + 2*cos(AOpc(POIo,t))*Rpch(POIo,t) ] where : Rpch(POIo,t) is at angle AOpc(POIo,t) RDEpdh(POIo,t,dt) is at angle Opda(RDEpdh(POIo,t,dt)) = AOpc(POI,t) + PI/2 This should be expected, as per the Figure "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo,t)]". #-----+ 22Jan2017 change Rpdh to RDEpdh? - makes it more clear & stands out RDEpdh(POIo(t)) is anchored at end of Rpch(POIo(t)) and is at angle AOpc(POIo(t)) - PI/2, ie perpendicular to Rpch(POIo(t)) >> done #-----+ YIKES!!! ALL Sections with d[dt : E] d[dt : B] are SCREWED UP!!! 22Jan2017 - mostly fixed, must go through all, should write Q'Nial check program #-----+ 9Jan2017 change ALL angle symbols to format Apco, Aocp, Aopv,Aocp0 where the last small letter [o,p,v] designates [theta, phi, general] additional (zero) denotes theta or phi angle of zero #-----+ 22Jan2017 In section "dp[dt : E0odv(POIo)] ~= dp[dt : E0pdv(POIp)] = 0" NYET -> use of "dp[dt : E0pdv(POIo(t),t)]" is WRONG = 0!!! - but only if POIo(t) is FIXED within RFp, which is NOT the significance of the symbol POIo(t). That (for Chapter 4) infers a particle fixed in the observer reference frame RFo, which is moving in RFp, hen ce POIo(t). This would also apply to a third reference frame (say POIx), for which POIo applies if that frame is not moving with respect to POIo. Key issue may be "ether-centric" field reference frame, given that relativity is WRONG by interferometer experiments (neither RFp nor RFo apply). Most likely affects many other sections as well 17Dec2017 I don't think that's correct, but I'm not sure as I've forgotten the nomenclature and specifics!! Look at where this occurs : save “Howell - Background math for Lucas Universal Force, Chapter 4.odt” as ASCII “text conversions/Howell - Background math for Lucas Universal Force, Chapter 4.txt” escapes for grep expressions with [?+{()] are trickey!!! don't escape [( ) ] in normal part of expressions $ grep --line-number "dp[[]dt \: E0pdv(POIo(t),t)" "/home/bill/Projects/Lucas - Universal Force/text conversions/Howell - Background math for Lucas Universal Force, Chapter 4.txt" +--+ 98:Differential vector geometry of dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)] 96 99:dp[dt : E0odv(POIo)] = dp[dt : E0pdv(POIo(t),t)] ~= dp[dt : E0odv(POIp(t))] = dp[dt : E0pdv(POIp)] = 0 97 3164:Differential vector geometry of dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)] 3191:Looking at Figure "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" : 3193:1) dp[dt : E0pdv(POIo(t),t)] 3220:Looking at Figure "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" : 3223: 1) dp[dt : E0pdv(POIo(t),t)] 3233:where, as in "Differential vector geometry of dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" : 3276: 1) dp[dt : E0pdv(POIo(t),t)] 3283:5) dp[dt : E0pdv(POIo(t),t)] 3295:From "Differential vector geometry of dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" : 3303:6) dp[dt : E0pdv(POIo(t),t)] 3330: 6) dp[dt : E0pdv(POIo(t),t)] 3392:9) dp[dt : E0pdv(POIo(t),t)] 3580:Note : This is the SAME as the Rpch component of " dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" : 3581: 6*) dp[dt : E0pdv(POIo(t),t)] 3587:This should be expected, as per the Figure "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]". 3807: 1) dp[dt : E0pdv(POIo(t),t)] 4564:From "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" : 4565: 6) dp[dt : E0pdv(POIo(t),t)] 4573: = -lambda(v)*dp[dt : E0pdv(POIo(t),t)] 4588:From "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" : 4589: 9)* dp[dt : E0pdv(POIo(t),t)] 4610: = -lambda(v)*dp[dt : E0pdv(POIo(t),t)] 4646: 2)* dp[dt : EI_LENZpdv(POIo(t))] = -lambda(v)*dp[dt : E0pdv(POIo(t),t)] 4846:Equation (4) is (1 - lambda(v)) times "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" Equation (6), as expected. 4980:Equation (5) is (1 - lambda(v)) times "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" Equation (9), as expected. 5829:1) = dp[dt : f_BARNES(Vonv(particle),AOpda(POIo(t)))]*E0pdv(POIo(t),t) + f_BARNES(Vonv(particle),AOpda(POIo(t)))*dp[dt : E0pdv(POIo(t),t)] 5869:4) f_BARNES(Vonv(particle),AOpda(POIo(t)))*dp[dt : E0pdv(POIo(t),t)] 5871:From "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" RFp basis : 5872: 6)* dp[dt : E0pdv(POIo(t),t)] 5881:5) f_BARNES(Vonv(particle),AOpda(POIo(t)))*dp[dt : E0pdv(POIo(t),t)] 5894:5) f_BARNES(Vonv(particle),AOpda(POIo(t)))*dp[dt : E0pdv(POIo(t),t)] 5910: + f_BARNES(Vonv(particle),AOpda(POIo(t))) *dp[dt : E0pdv(POIo(t),t)] 6005:From "dp[dt : E0odv(POIo,t)] = dp[dt : E0pdv(POIo(t),t)]" RFp basis : 6006: 9) dp[dt : E0pdv(POIo(t),t)] +--+ #-----+ 22Jan2017 dp[dt : Rpch(POIo(t)) ] -> and is at angle AOpc(POIo(t)) - PI/2, ie perpendicular to Rpch(POIo(t)) shouldn't this be PLUS? AOpc(POIo(t)) + PI/2, 19Dec2017 see Figure "Basic measures for for the particle reference frame RFp, using POIp=POIo(tx) Correct - when Vonv(particle is positive, then delta{AOpc(POIo(t))} is positive, and delta{AOoc(POIp(t))} is negative $ grep --line-number "\- PI/2" "/media/bill/Lexar/Lucas - Universal Force/text conversions/Howell - Background math for Lucas Universal Force, Chapter 4.txt" +--+ 999:Distance of R_OPI2_pcs(POIp) = R_OPI2_ocs(POIo) from L(particle) in Ppch=Poch direction (i.e. perpendicular to L(particle)). Note that the notation "_OPI2_" is a mnemonic for theta = O (capital O) = PI/2 radians. 1322: = Vons(particle)*d[t]*cos(AOpc(POIo(t)) + PI/2) ...where d[theta] is a differential change in O 1323:But : cos(alpha) = sin(alpha + PI/2) 1387:As AOpc(POIo(t)) -> PI/2 : 1425: and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 1441: and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 23Jan2017 changed to PLUS PI/2? (was that way in half of the expressions - maybe should be for RFp? 1473: and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 3174:1a) AOpda(RDEpdh(POIo(t),dt)) is the direction perpendicular to Rpch(POIo(t)), rotated in the positive Opna (or Oona) direction (i.e at an angle of PI/2 from Rpch(POIo(t))). This gives an angle with respect to the (RFo) theta coordinate vector, AOpc(POI,t), of (PI/2). AOpda(RDEpdh(POIo(t),dt)) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? 3176: AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 3187: = AO_E0DIFF_oca(POIo(t),dt) + PI/2 + AOpc(POIo(t)) 3234: 1a) AOpda(RDEpdh(POIo(t),dt)) is the direction perpendicular to Rpch(POIo(t)), rotated in the positive Opna (or Oona) direction (i.e at an angle of PI/2 from Rpch(POIo(t))). This gives an angle with respect to the (RFo) theta coordinate vector of (PI/2). AOpda(RDEpdh(POIo(t),dt)) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? 3296: 1a) AOpda(RDEpdh(POIo(t),dt)) is the direction perpendicular to Rpch(POIo(t)), rotated in the positive Opna (or Oona) direction (i.e at an angle of PI/2 from Rpch(POIo(t))). This gives an angle with respect to the (RFo) theta coordinate vector, AOpc(POI,t), of (PI/2). AOpda(RDEpdh(POIo(t),dt)) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? 3297: AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 3308: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POIo(t)) + PI/2 3335: AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 3370: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 3400: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 3469: and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 3536: RDEpdh(POIo(t)) is anchored at end of Rpch(POIo(t)) and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 3539: = |Rpch(POIo(t))| * |RDEpdh(POIo(t))|*cos(AOpc(POIo(t)) + PI/2 - AOpc(POIo(t))) 3540: = |Rpch(POIo(t))| * |RDEpdh(POIo(t))|*cos(PI/2) 3586: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 3717:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EIpds(POIo(t))] < 0 3718:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EIpds(POIo(t))] > 0 3772:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EIpds(POIo(t))] < 0 3773:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EIpds(POIo(t))] > 0 3842: AngleBetween(plane of Vonv & Rpcv(POIo(t)), APod(BTodv(POIo,t))) = constant = PI/2 in phi (i.e. APpda(V X R)) 3843: APoc(BT(POIo(t))) = constant = APoc(POIo) + PI/2 (again assuming BI,BT are collinear) 4381:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EI_LENZpds(POIo(t))] < 0 4382:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EI_LENZpds(POIo(t))] > 0 4570: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 4580: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 4597: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 4636: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 4651: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 4675:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EI_LENZpds(POIo(t))] > 0 4676:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EI_LENZpds(POIo(t))] < 0 4799: RDEpdh(POIo(t)) is anchored at end of Rpch(POIo(t)) and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 4872: and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 4967: and is at angle AOpc(POIo(t)) + PI/2, ie perpendicular to Rpch(POIo(t)) 5877: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 5887: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 5939: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 6014: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 6096: RDEpdh(POIo(t),dt) is at angle AOpda(RDEpdh(POIo(t),dt)) = AOpc(POI,t) + PI/2 +--+ >> OK - almost all are "+ PI/2" >> check : 3717:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EIpds(POIo(t))] < 0 3718:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EIpds(POIo(t))] > 0 3772:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EIpds(POIo(t))] < 0 3773:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EIpds(POIo(t))] > 0 4381:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EI_LENZpds(POIo(t))] < 0 4382:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EI_LENZpds(POIo(t))] > 0 4675:- -PI/2 <= AOpc(POIo(t))) <= PI/2 : then cos(AOpc(POIo(t))) >0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EI_LENZpds(POIo(t))] > 0 4676:- PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] < 0, and dp[dt : EI_LENZpds(POIo(t))] < 0 >> I added phrasing (example for last two) 19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? - PI/2 <= AOpc(POIo(t))) <= -PI/2 : then cos(AOpc(POIo(t))) <0 and dp[dt : E0pds(POIo(t),t)] > 0, and dp[dt : EIpds(POIo(t))] > 0 30Mar2018 put Lucas book derivations in same odt format!! enddoc