Table of Contents
Changes required 1
Main file “Howell - math of Lucas Universal Force.ndf” 1
Backlog file "Howell - math of Lucas Universal Force.ndf" : 3
Old listings ... 4
Work in progress 6
III.5 Summary comparison of measures and derivatives (FRp) versus (FRo) 6
Galilean transformation FRp <=> FRo 6
06Apr2016 Figures ToDo 6
Scalar absolute values, [vector, matrix] norms - simplification of expressions 8
New set 8
Completed 10
NUTS!! Improper, indefinite integrals - most results only have sin terms, in which case the lower result = 0. 11
16Jun2016 phfrm Lucas :
incl problems with scientists, socialist education system
- EM waves [longitudinal - life, no travel time; transverse - speed of c, gravity]
Changes required
Main file “Howell - math of Lucas Universal Force.ndf”
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)
#-----+
#-----+
?date? check binomial series coefficients 4-37 through 4-46
19Dec2017 see "Binomial series" sub-section of "Howell - Background math for Lucas Universal Force, Chapter 4.odt"
Third iteration ((4-37a))
Substitute LHS ((4-38)) into ((4-32))
#-------+
Don't induced E & B chage direction every other iteration?
Backlog file "Howell - math of Lucas Universal Force.ndf" :
12Jun2016 Iterate (4-34) in same manner as (4-33).
But isn't that what I am doing? But the dropping of (1 - cosO) is obccure, as is
08Jun2016 Redo (4-34) !!! I had different result, and used it for (4-37), but I improved my integrals/derivatives since.
But this will ONLYsolve a small part of the problem!
I actually need to redo (4-32) through (4-3&)!!! TOO MUCH TIME!!
+----->
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)
<-----+
Dimensions (units) for F don't balance - see "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt"
- also there is an imbalance in (4-34) - see "Howell - math of Lucas Universal Force.ndf"
my error with integrals & derivatives!!
- equations ??? many of them!
- especially Thomas Barnes iterations
Discussions for key equations - I have simply gone through the MECHANICS of the derivations, with only occasional comments with regards to their basis. For example :
• (4-29) symmetry of EIods is not clear - this is theta-dependent, but here it is the wording perhaps that is confusing me.
• (4-30) line integral of reformulated E to take place of B.
Old listings ...
17May2016 equation (4-1) - for Chapter 4 situation - perhaps the curl resolves to one dimension ONLY??? Still would yield derivative of B !!!
17May2016 HOWEVER, I still have to go through Appendix A for the Generalized Ampere's Law.
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) !!!
17Mar2016 finite-size particle - I have to create new sub-sections and provide Lucas' forms of equations, as the point-size particle approach may not get there. As a start, just use an electron [toroid, plasmoid, monad, donut]
25Mar2016 For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the derivations (not as straightforward as it sounds!).
(Notably : dp[dt : E0ods(POIo,t)], dp[dt : EIods(POIo,t)], dp[dt : ETods(POIo,t)], dp[dt : BTods(POIo,t)] etc)
for example : dp[dt : EIods(POIo,t)] has the opposite sign of dp[dt : E0ods(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 Derivatives in spherical coordiantes - I have done the initial symbolic programming to work with this, but have not yet "closed the loop" between the standard formulation and its application to Lucas's Chapter 4. RFo,RFp, POIo, POIp, Vonv etc. I must be able to formally derive a few example derivatives that I have done by more direct approaches in this document.
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)
31Mar2016 No time delays accounted for in electrodynamic equations
16May2016 "Howell - Background math for Lucas Universal Force, Chapter 4.odt"
I changed symbols for angles, eg Ooca -> AOoc, Opca -> AOpc, others Ooda, Ppca, Poca, Poda
But this still has to be checked as errors will have been made with search & replace
Also - must change angle symbols in :
- Figures
- Howell - math of Lucas Universal Force.ndf
- 0_Math symbols.odt
- Howell - Background math, summary listing of Chapter 4 formulae.odt
- other files???
31Mar2016 Big questions at the base :
Normally, Maxwell's equations relate RATE OF CHANGE of B to E : as with (4-2) Faraday'sd Law, but not like (4-1) Generalized Ampere's Law (actulally latter probably OK?)
Lenz's Law - is KEY, need proof from other approaches as this is a HUGE simplification!
Barnes iterations - how does this look fundamentally-geometrically?
01Apr2016 In (4-43) is theta Opca, Ooca, or what? This affects derivatives of f_BARNES!!!
04May2016 NET, this is not correct -> 12May2016 p68h0.5 Figure (4-4) Halelujeh!! -- Sometimes, Lucas's angle theta is the angle at a point on the surface of a charged particle between the suface normal vector and B !!!
12May2016 I should start all sysmbols for angles with "A" - eg "AOoc(POIo)" instead of "Ooca(POIo)"; and "APpc(POIo,t)" instead of "Ppca(POIo,t)". In other words, the small case "a" in the modifier list moves to the front of the symbol. Here ["R","O","P"] are like "X","Y","Z" for Cartesian coordinates. "A(PVox,Rodv_start,Rodv_end)" is a general way of defining an angle, where the modifier lists labelled "o" and "odv" here can have either [o,p] for the first modifier character, but must have "dv" to denote [anchored, directed] vectors starting from a common "vertex" (point in space). The modifier "x" indicates the point that is the "vertex" of an angle.
26May2016 Partial versus total derivatives - I've simpky used partial derivative notation for almost every expression, rather thna properly using total derivative notation as I should have in some cases (eg dp[dt : EIodv(POIo,t)] rather than dT[dt : EIodv(POIo,t)].
#-----+
12Oct2015 - Put to the side for 3 weeks!!!
Next - as per "Howell - Key math info & derivations for Lucas Universal Force.odt" subsection II.3 "Expression for dp[dt : E0(POI,t) ) - Spherical from Cartesian first principles approach" :
However, (4-15) and (4-16) are expressions for B(POI,t), which involves a CROSS-PRODUCT of E(POI,t). Therefore, Pp(POI,t) comes into play, and the real comparison I need to make is with an expression for B(POI,t).
#-----+
# 02Oct2015 as per (4-37rev9), there are several basic problems with
Luca's writeup. Perhaps this not so much a problem with the underlying
concepts and work before writeup (as he has had problems with dropped
symbols etc in his presentations).
# Next step - "fudge" a guess at the appropriate (4-34) based on my
(Howell's) own results for earlier equations, and use that to re-derive (4-37).
I want to see if I get "somewhat close" to what I should get so I don't
waste a huge amount of more time.
# 05Jan2016 - NOOOO!! this is incorrect!! :
# After that plan - redo all derivations involving |ro - vo*t| (change to |ro|),
and use my (Howell's) results instead of Lucas.
This targets most, if not all, of : (4-15), (4-16), (4-27), (4-30), (4-32) to (4-37)
Because this is so much work, it's best for me to "fudge" a guess at
the appropriate (4-34), and use that to re-derive (4-37). I'm worried
that there are still hidden problems that would render a rework useless.
#-----+
# 23Sep2015 - I have to mark carefully the scalars, eg rs -> use rs
# Problems with sin,cos - (4-34), 04_41, 04_42,
(4-34)rev2 CLOSE - middle term RHS : [mine, Lucas] for [, cosO]
(4-35rev2 CLOSE - [Howell,Lucas] [-1,1], Howell has extra r"s in two terms?
(4-37rev2 NUTS - looks like I misclassified a L(v) term]
(4-41 PROBLEM - sin,cos terms
# Lesser problems
4-19 WRONG - Notice that I didnt have to apply Stokes theorem!
4-24 WRONG - On hold as I havent been able to "move" (v/c) to the right place..
4-28a OK - but note that I have taken 1/r out of the [] for Oh
4-29a WRONG - Lucas expression relates to ∇Ei(ro - vo*t,t) and NOT
#-----+
# 25Aug2015 Must verify Appendix A Equation (A19)
# 23Aug2015 Appendix A - historical basis to Maxwell's equations is scarily
"fluffy" - a lot of great math simplifications, but what about the
experimental confirmation?
endsection
Work in progress
III.5 Summary comparison of measures and derivatives (FRp) versus (FRo)
Context for (RFp), (POIp)
Galilean transformation FRp <=> FRo
Rocv(POIp,t) = Rpcv(POIp) + Vonv(particle)*t
Rocs(POIp,t))
= { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Opca(POIp))*Vons(particle)*t + [Vons(particle)*t]^2 }^(1/2)
ROPI2ocs(POIp,t) = ROPI2pcs(POIp) = Rpcs(POIp)*sin(Opca(POIp))
sin(Ooca(POIp,t))
= Rpcs(POIp)*sin(Opca(POIp))
/{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Opca(POIp))*Vons(particle)*t + [Vons(particle)*t]^2 }^(1/2)
)a(POIo,t)
Rocv(POIo)
Rocs(POIo)
R_OPI2_ocs(POIo) = R_OPI2_pcs(POIo)
R_O0_ocs(POIo)
R_O0_pcs(POIo,t)
Rpcv(POIo,t)
Rpcs(POIo,t)
sin(Opca(POIo,t))
cos(Opca(POIo,t))
06Apr2016 Figures ToDo
14Jun2016 IV. Chapter 4 - Expressions for the t=0 reference frame (RFt)
Galilean transformmation
III.1 IntRoduction & (Observer, particle) reference frames : ??? show Figure of "capped" sphere ???
dp[dt : Rocs(POIp,t) ] = dp[dt : |Rocv(POIp,t)|] : Show Figure "dp[dt : Rocs(POIp,t)] explained"
??? maybe not?? dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)] - cheating E0ods(POIo,t) scalar approach : see Figure "dp[dt : E0odv(POIo,t)] - basic metrics"
#-----+
# 22Sep2015 1st try, 23Sep2015 rev2, 26Sep2015 rev3&4, 27Sep2015 rev5
# 27Sep2015 rev7, 29Sep15 rev8, 30Sep2015 rev9, 02Oct2015 no workee, 02Oct2015 rev10
# 05Jan2016 rev11 use dp[dt : |ro - vo*t|] = 0, "fudge" a guess at the
# appropriate (4-34) based on my (Howell's) own results
# for earlier equations
# 27May2016 ??? What did I do with the derivations!!??? - looks like I didn't get around to the next try.
# see "Howell - Old math of Lucas Universal Force.ndf"
# 29May2016 rev12 using Howell's "finalized" (4-34)rev5 29May2016
# 05Jun2016 rev13 implementing better integral expressions for ∫{dAOtc, 0 to AOpcf : cos(AOpc(POIo,t))*sin^a(AOpc(POIo,t))*Rpcs(POIo,t)^(-b) }
# 06Jun2016 rev14 cancel out Rocs,Rpcs after each integration - even bring "outside " Rocs into integral!!
# 08Jun2016 rev15 redo with "improved" nomenclature
# 10Jun2016 rev16 This version "unjustifiably" drops an extra term from (4-34)rev5 to get (4-37)rev16, as was the case with (4-32)rev5 to get (4-33)rev1.
# 10Jun2016 rev17 Direct substitution of 4-36 into 4-34 (handles dropping of v*t*(1 - cosO)), NOTE: I tried E0ods(POIo,t=0) as constant for integration & derivatives just for fun - no workee
# 13Jun2016 rev18 Direct substitution of 4-36 into 4-32 (handles dropping of v*t*(1 - cosO))
# 22Sep2015 1st try,
# 23Sep2015 rev2,
# 26Sep2015 rev3&4
# 27Sep2015 rev5 ∫(dOp, 0 to Of : sinOp *(rs*cosOp - vs*t) )
# 27Sep2015 rev7 ∫(dOp, 0 to Of : sin^3(Op)*cosOp )
# 30Sep2015 rev8 ∫(dOp, 0 to Of : sin^3(Op)*cosOp )
# 02Oct2015 rev9 ∫(dOp, 0 to Opf : sin^3Op*cosOp )
# 29May2016 rev12 ∫{dAOpc, 0 to AOpcf : sin(AOpc(POIo,t))sin^2(AOpc(POIo,t))*cos(AOpc(POIo,t)) }
# 05Jun2016 rev13 ∫{dAOpc, 0 to AOpcf : sin(AOpc(POIo,t))sin^2(AOpc(POIo,t))*cos(AOpc(POIo,t))*Rpcs(POIo,t)^(-7) }
# 06Jun2016 rev14 ∫{dAOtc, 0 to AOpc(POIo,t=0) : cos(AOtc)*sin(AOtc )^3 }
# 08Jun2016 rev15 ∫{dAOtc, 0 to AOpc(POIo,t=0) : sin(AOtc )*cos(AOtc)*sin(AOtc )^2 }
# 10Jun2016 rev16 ∫{dAOtc, 0 to AOpc(POIo,t=0) : sin(AOtc(RFt))^1*cos(AOtc(RFt))*sin(AOtc(RFt))^2 }
# 10Jun2016 rev17 ∫{dAOtc, 0 to AOpc(POIo,t=0) : sin(AOtc(RFt))^1*cos(AOtc(RFt))*sin(AOtc(RFt))^8 }
# 13Jun2016 rev18 Direct substitution of 4-36 into 4-32 (handles dropping of v*t*(1 - cosO))
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,...)| <> != product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases)
vector norms - if some terms are vectors
- multiplicative/divisive : |product(x1,x2,x3,...)| <> != product(|x1|,|x2|,|x3|,...)
- subtraction/ addition : |sum (x1,x2,x3,...)| <> != product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases)
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
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
enddoc