www.BillHowell.ca's review of Bill Lucas's math in his book the "Universal Force, Volume I", equationS (4-[32-37])
/*$ echo "version= $date_ymdhm, cos - 1 inclusion : $cos_inclusion" >>"$p_augmented"
This file is :
/*$ echo " $p_augmented" >>"$p_augmented"
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SUMMARY
This file is a focus on the effects on Lucas's equations (4-32) to (4-37) of one of several different treatments of
∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] :
- included in derivations
- arbitrarily set to zero
/*$ cat >>"$p_augmented" "$d_Lucas""context/d-dt Rpcs^-5*t*_cos - 1 comments.txt"
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/*$ cat >>"$p_augmented" "$d_Lucas""context/text editor - how to set up.txt"
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/*------> ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]
2015-2016 earlier versions
16Oct2019 clear rational and results for whether or not this term should be included in derivations
/*$ cat >>"$p_augmented" "$d_Lucas""math Howell/cos - 1 $cos_inclusion, iterative, non-feedback/d-dt Rpcs^-5*t*_cos - 1.txt"
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/*------> (4-32) EIods(POIo,t=0,1st stage), F therefore E balance - iteration #1 on (4-30)
initial - 15Sep2015, rev1 - ??, rev2 - 25Sep2015
15Sep2015 p71h0.33 Iterate [(4-31) into (4-30)] to obtain solution.
09Jun2016rev5 use HFLN, corrected (I hope) [reference frames, notations, derivatives, integrals]
14Jun2016rev6 fix my error (cosOp - 1) !!
22Aug2019 remove (cosOp - 1) term
30Sep2019 corrections, cleanup
/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_32 work.txt"
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/*------> (4-33) EIods(POIo,t=0, 2nd stage), F therefore E balance
29May2016 I had mistakenly used Ei in place of Eis, so this has been corrected in the final result.
14Jun2016rev3 fix my error (cosOp - 1) !!
Lucas"s version of (4-32)
(4-32rev3 F therefore E balance - simplified (4-30)
22Aug2019 start revision used revamped (4-32), finished 26Aug2019
03Sep2019 fix error with Equation (5) below - dropped a term, K3 was incorrect then simply dropped from expressions
24Sep2019 drop K1, add "Highly restrictive conditions"
30Sep2019 correct + sign for K1
/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_33 work.txt"
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/*------> (4-34) EIods(POIo,t=0, 2nd stage), K_2nd from taking partial derivatives wrt time
for previous versions, see "Howell - Old math of Lucas Universal Force.ndf"
16Sep2015 1st or 2nd?, 20Sep2015 rev2, 25Sep2015 rev4
29May2016rev5
14May2016rev6
13Aug2019 !!! ∂[∂(x): sin(x)] = cos(x) !!! ????????????????????????? /%
27Aug2019 revamp, hopefully fixing wrong [+,-] issues for (4-37)
There is an ambiguity in my expressions with [t, t=0] - sometimes I've left in the wrong form
03Sep2019 fix error in K2 term from 04_33 (actually - this was not a problem?)
03Sep2019 Note that I corrected the results there, which had Aθpc(POIp(t),t), which is an illegal symbol!!!
this was corrected to Aθoc(POIp(t),t) or Aθpc(POIo(t),t)
03Sep2019 no net change to end result of 04_34
24Sep2019 following changes to 4-33, drop K1, add "Highly restrictive conditions"
02Oct2019 fix ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a], add K_1st term
14Oct2019 fix generative form for 4-34 (forgot to do that earlier)
/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_34 work.txt"
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/*------> (4-35) E0ods(POIo,t) truncated expression with ONLY E0ods(POIo,t) terms
16Sep2015 1st version, ?17Sep2015? rev1, 21Sep2015 rev2, 26Sep2015 rev3
for previous versions, see "Howell - Old math of Lucas Universal Force.ndf"
??? 04Jan2016 Check if should be redone with "Howell - Key math info & derivations for Lucas Universal Force.odt"
29Aug2019 re-derive based on recent work from (4-32) to (4-34)
03Sep2019 fix ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIp(t),t)) ] :
I had sin(Aθpc(POIo(t),t=0))^1/1, should have been sin(Aθpc(POIo(t),t=0))^2/2
24Sep2019 following changes to 4-33, drop K1, add "Highly restrictive conditions"
02Oct2019 percolate fix to ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a], note K_2nd term
/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_35 work.txt"
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/*------> (4-36) ETods(POIo,t) expression with ONLY ETods(POIo,t) terms
17Sep2015 2nd iteration
03Sep2019 Redone with HFLN - it now works!!!!!
/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_36 work.txt"
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/*------> (4-37) Er and the binomial series, leading to the relativistic correction factor
27Sep2019 - did not finish - revamp in next attempt with updated previous equations 4-[32 to 36]
27Sep2019 using updated previous equations 4-[32 to 36]
02Oct2019 add [K_1st, K_2nd] terms
after fixing ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a]
fix ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a]
11Oct2019 use corrected form of EIods(POIo,t,2nd stage) with "+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}"
Approach for 22Aug2019, 03Sep2019 after correcting recent errors in 04_35 :
1. Don't track Lucas too closely - he has wrong coefficients
2. Iterations pass through same operations as (4-32) through (4-36)
3. Howells "dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)] - see proper E0odv(POIo,t) vector approach" from "Howell - Background math for Lucas Universal Force, Chapter 4.odt"
I was very lazy with symbols -0 many improper uses!!! Must clean up thre t=0 notational mess for [integrals, derivatives]
/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_37 work.txt"
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enddoc