DUAL_HYPERQUATERNION_POINCARE_GROUPS.nb
Author
Patrick R. Girard, Patrick Clarysse, Romaric Pujol, Robert Goutte, Philippe Delachartre
Title
DUAL_HYPERQUATERNION_POINCARE_GROUPS.nb
Description
Supplemental notebook to "Dual Hyperquaternion Poincaré Groups"
Category
Academic Articles & Supplements
Keywords
quaternions, hyperquaternions, dual hyperquaternions, Poincaré groups, canonical decomposition
URL
http://www.notebookarchive.org/2021-03-4mrwcdg/
DOI
https://notebookarchive.org/2021-03-4mrwcdg
Date Added
2021-03-10
Date Last Modified
2021-03-10
File Size
373.48 kilobytes
Supplements
Rights
Redistribution rights reserved
Download
Open in Wolfram Cloud
In[]:=
(*WRITTENANDTESTEDWITHMATHEMATICA8FORTHERESEARCHPAPER:GIRARD,P.R.,CLARYSSE,P.,PUJOL,R.,GOUTTE,R.,DELACHARTRE,P.,DUALHYPERQUATERNIONPOINCAREGROUPS,Adv.Appl.CliffordAlgebras31,15(2021).https://doi.org/10.1007/s00006-021-01120-z*)(*******************************************************)<<Quaternions`Remove["Global`*"]
In[]:=
(*------------------------------------------------------*)(*5.1.ALGEBRAICFORMULATION,EQUATION(5.1)*)(*APPENDIXC:MULTIVECTORSTRUCTUREOF⊗⊗,EQUATIONS(C.1,C.2)*)(*HyperquaternionAlgebra⊗⊗,APPENDIXC,EQUATION(C.1)*)(*AlgebraicIdentification:A=(a1+l*a2+m*a3+n*a4)+(I*a5+Il*a6+Im*a7+In*a8)+(J*a9+Jl*a10+Jm*a11+Jn*a12)+(Ka13+Kl*a14+Km*a15+Kn*a16)+(i*a17+il*a18+im*a19+in*a20)+(iI*a21+iIl*a22+iIm*a23+iIn*a24)+(iJa25+iJl*a26+iJm*a27+iJn*a28)+(iK*a29+iKl*a30+iKm*a31+iKn*a32);aμν:realnumbers.A:={Quaternion[a1,a2,a3,a4],Quaternion[a5,a6,a7,a8],Quaternion[a9,a10,a11,a12],Quaternion[a13,a14,a15,a16],Quaternion[a17,a18,a19,a20],Quaternion[a21,a22,a23,a24],Quaternion[a25,a26,a27,a28],Quaternion[a29,a30,a31,a32]}*)(*Examples,APPENDIXC,EQUATIONS(C.1-C.2),TABLE*)(*l:={Quaternion[0,1,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}m:={Quaternion[0,0,1,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}n:={Quaternion[0,0,0,1],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}I:={Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}J:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}K:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}i:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}i*J*m:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,1,0],Quaternion[0,0,0,0]}e0(=i*J):={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[0,0,0,0]}e1(=i*K*l):={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,1,0,0]}e2(=i*K*m):={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,1,0]}e3(=i*K*n):={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,1]}*)
In[]:=
In[]:=
(*Conjugate*)K[a_]:={Quaternion[a[[1,1]],-a[[1,2]],-a[[1,3]],-a[[1,4]]],Quaternion[a[[2,1]],-a[[2,2]],-a[[2,3]],-a[[2,4]]],Quaternion[a[[3,1]],-a[[3,2]],-a[[3,3]],-a[[3,4]]],Quaternion[-a[[4,1]],a[[4,2]],a[[4,3]],a[[4,4]]],Quaternion[-a[[5,1]],a[[5,2]],a[[5,3]],a[[5,4]]],Quaternion[-a[[6,1]],a[[6,2]],a[[6,3]],a[[6,4]]],Quaternion[-a[[7,1]],a[[7,2]],a[[7,3]],a[[7,4]]],Quaternion[a[[8,1]],-a[[8,2]],-a[[8,3]],-a[[8,4]]]}
In[]:=
(*Dual*)II:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}dual[a_]:=HP[II,a]
In[]:=
(*Multivectorextraction:scalars,bivectors,trivectors,...*)V0[a_]:={Quaternion[a[[1,1]],0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}V1[a_]:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[a[[7,1]],0,0,0],Quaternion[0,a[[8,2]],a[[8,3]],a[[8,4]]]}V2[a_]:={Quaternion[0,a[[1,2]],a[[1,3]],a[[1,4]]],Quaternion[0,a[[2,2]],a[[2,3]],a[[2,4]]],Quaternion[0,a[[3,2]],a[[3,3]],a[[3,4]]],Quaternion[a[[4,1]],0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}V3[a_]:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,a[[5,2]],a[[5,3]],a[[5,4]]],Quaternion[0,a[[6,2]],a[[6,3]],a[[6,4]]],Quaternion[0,a[[7,2]],a[[7,3]],a[[7,4]]],Quaternion[a[[8,1]],0,0,0]}V4[a_]:={Quaternion[0,0,0,0],Quaternion[a[[2,1]],0,0,0],Quaternion[a[[3,1]],0,0,0],Quaternion[0,a[[4,2]],a[[4,3]],a[[4,4]]],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
In[]:=
(*ParticularHyperquaternions*)zero:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}U:={Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}miI:={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[-1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}e0123:={Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
In[]:=
(*-------------------------------------------------------*)(*5.EXAMPLE:4DPOINCAREGROUP*)(*5.1ALGEBRAICFORMULATION,EQUATION(5.6)*)X={Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[ϵ*xx0,0,0,0],Quaternion[0,ϵ*xx1,ϵ*xx2,ϵ*xx3]}
Out[]=
{Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[xx0ϵ,0,0,0],Quaternion[0,xx1ϵ,xx2ϵ,xx3ϵ]}
In[]:=
(*------------------------------------------------------*)(*5.2NUMERICALEXAMPLE:EQUATION(5.9)*)(*Spatialrotation*)A1={Quaternion[(Sqrt[3])/2,0,1/2,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}X'=Chop[Simplify[HP[A1,HP[X,K[A1]]]]](*Space-timetranslation*)A2={Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,-2*ϵ,0,0],Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}X'=Chop[Simplify[HP[A2,HP[X,K[A2]]]]](*Hyperbolicrotation:boost*)A3={Quaternion[2,0,0,0],Quaternion[0,0,Sqrt[3],0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}X'=Chop[Simplify[HP[A3,HP[X,K[A3]]]]](*ComputationoftheproductA3*A2*A1=f(=a),equation(5.9)*)a=Simplify[HP[A3,HP[A2,A1]]]X':=Chop[Simplify[HP[a,HP[X,K[a]]]]]
Out[]=
Quaternion,0,,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
3
2
1
2
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion[xx0ϵ,0,0,0],Quaternion0,(xx1+(-
1
2
3
xx3)ϵ,xx2ϵ,1
2
3
xx1+xx3)ϵOut[]=
{Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,-2ϵ,0,0],Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
Out[]=
{Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1+(3-2xx0+4xx1),0,0,0],Quaternion[ϵ(2+xx0-5xx0+4xx1),0,0,0],Quaternion[0,ϵ(4+xx1-4xx0+5xx1),xx2(ϵ-3),xx3(ϵ-3)]}
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
3
ϵ
3
ϵ
Out[]=
Quaternion[2,0,0,0],Quaternion0,0,
3
,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion7xx0ϵ+4
3
xx2ϵ,0,0,0,Quaternion0,xx1ϵ,43
xx0ϵ+7xx2ϵ,xx3ϵOut[]=
Quaternion,0,,0,Quaternion,-2,-2ϵ,Quaternion
3
,0,1,0,Quaternion-3
2
3
2
3
ϵ2
3
ϵ,-3ϵ
2
3
ϵ,-3
ϵ,ϵ,3ϵ,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]In[]:=
(*-------------------------------------------------------*)(*5.2NUMERICALEXAMPLE:EQUATION(5.10)*)(*Canonicaldecomposition,determinationofb1,b2,B1,B2,ϕ1,ϕ2;REF.[9]*)(*PrincipleandBasicequations:P1=B=X1+X2,[a]P2=(B∧B).B=2(x1*X2+x2*X2),[b]S1=P1.P1=x1+x2,[c]S2=P2.P1=4x1.x2,[d]withX1=b1B1,X2=b2B2,x1=X1^2,x2=X2^2,Onesolvesx1,x2viatheequations[c]and[d]andX1,X2intermsofP1,P2,x1,x2viaequations[a]and[b]*)(*ComputationofthebivectorP1=B*)S=a[[1,1]]P1=Simplify[fclif[1/S,V2[a]]]B=P1(*ComputationofthebivectorP2*)BextB=Simplify[V4[HP[B,B]]]P2=Simplify[V2[HP[BextB,B]]](*ComputationofthescalarsS1andS2*)P1intP1=Chop[Simplify[HP[P1,P1]]]S1=P1intP1[[1,1]]/.ϵ_^n_0P2intP1=Chop[Simplify[HP[P2,P1]]]S2=P2intP1[[1,1]]/.ϵ_^n_0(*Determinationofx1,x2viatheequations[c]and[d]*)s=Chop[Simplify[Solve[{x1+x2S1,4*x1*x2S2},{x1,x2}]]]xx=s[[All,1,2]]yy=s[[All,2,2]]x1=xx[[1]]x2=yy[[1]](*DeterminationofX1,X2viathelinearequations[a]and[b];Solution:X1=p*P1+q*P2,X2=r*P1+s*P2;determinationofp,q,s,r*)P1P2uu=Simplify[Solve[{X1+X2PP1,2*x2*X1+2*x1*X2PP2},{X1,X2}]]xxx=uu[[All,1,2]]yyy=uu[[All,2,2]]xxx[[1]]PP1=1PP2=0p=xxx[[1]]PP1=0PP2=1q=xxx[[1]]yyy[[1]]PP1=1PP2=0r=yyy[[1]]PP1=0PP2=1s=yyy[[1]]X1A=fclif[p,P1]X1B=fclif[q,P2]X1=Chop[Simplify[csum[X1A,X1B]]]X2A=fclif[r,P1]X2B=fclif[s,P2]X2=Chop[Simplify[csum[X2A,X2B]]]X1X2b1=Sqrt[Abs[x1]]b2=Sqrt[Abs[x2]]B1=If[b1>0,Chop[fclif[1/b1,X1]],zero]B2=If[b2>0,Chop[fclif[1/b2,X2]],zero]b1b2B1B2ϕ1=2*ArcTan[b1]//Nϕ2=2*ArcTanh[b2]//N(*Verifications1:ofthesimpleplanesB1,B2andtheircommutativity*)w=Chop[Simplify[HP[B1,B1]]]/.ϵ_^n_0w=Chop[Simplify[HP[B2,B2]]]/.ϵ_^n_0w=Chop[Simplify[ext[B1,B2]]]/.ϵ_^n_0(*Verifications2:reconstructionofaandcomparisonwithinitiala*)Ch=Cosh[ϕ2/2]CC=Cos[ϕ1/2](*XX1=1+X1,XX2=1+X2*)XX1=Chop[csum[U,X1]]XX2=Chop[csum[U,X2]]aaa=a//Naa=Chop[fclif[CC*Ch,HP[XX1,XX2]]]//Nwww=Chop[Simplify[cdif[aa,aaa]]]www=Chop[Simplify[cdif[aa,aaa]]]/.ϵ_^n_0
Out[]=
3
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-2ϵ,-,-,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
3
ϵ2
2ϵ
3
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-2ϵ,-,-,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
3
ϵ2
2ϵ
3
Out[]=
Quaternion[0,0,0,0],Quaternion[-1,0,0,0],Quaternion[ϵ,0,0,0],Quaternion0,-2ϵ,,2
2ϵ
3
3
ϵ,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]Out[]=
Quaternion0,4,-,-,Quaternion0,-,,-,Quaternion0,-3ϵ,,-,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
2
ϵ
-3+
2
ϵ
2
3
4
2
ϵ
3
5
2
ϵ
3
-1+16
2
ϵ
3
7
2
ϵ
3
ϵ
3
3
ϵ,Quaternion-2ϵ
3
Out[]=
Quaternion(5+61),0,0,0,Quaternion[-1,0,0,0],Quaternion[ϵ,0,0,0],Quaternion0,-2ϵ,,2
1
12
2
ϵ
2ϵ
3
3
ϵ,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]Out[]=
5
12
Out[]=
Quaternion-1+,0,0,0,Quaternion(-5-61),0,0,0,Quaternionϵ(5+61),0,0,0,Quaternion0,-ϵ(5+61),,,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
49
2
ϵ
3
1
12
2
ϵ
1
12
2
ϵ
1
6
2
ϵ
ϵ(5+61)
2
ϵ
6
3
ϵ(5+61)
2
ϵ
2
3
Out[]=
-1
Out[]=
x1-,x2,x1,x2-
1
3
3
4
3
4
1
3
Out[]=
-,
1
3
3
4
Out[]=
,-
3
4
1
3
Out[]=
-
1
3
Out[]=
3
4
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-2ϵ,-,-,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
3
ϵ2
2ϵ
3
Out[]=
Quaternion0,4,-,-,Quaternion0,-,,-,Quaternion0,-3ϵ,,-,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
2
ϵ
-3+
2
ϵ
2
3
4
2
ϵ
3
5
2
ϵ
3
-1+16
2
ϵ
3
7
2
ϵ
3
ϵ
3
3
ϵ,Quaternion-2ϵ
3
Out[]=
X1(2PP1+3PP2),X2(3PP1-2PP2)
2
13
3
13
Out[]=
(2PP1+3PP2)
2
13
Out[]=
(3PP1-2PP2)
3
13
Out[]=
2
13
Out[]=
1
Out[]=
0
Out[]=
4
13
Out[]=
0
Out[]=
1
Out[]=
6
13
Out[]=
-
6
13
Out[]=
1
Out[]=
0
Out[]=
9
13
Out[]=
0
Out[]=
1
Out[]=
-
6
13
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-,-,-,Quaternion,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
4
13
3
2
3
13
8ϵ
13
2
3
ϵ13
8ϵ
13
3
4ϵ
13
Out[]=
Quaternion0,,-),-,Quaternion0,-,),-,Quaternion0,-,,-,Quaternion-,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
1
13
3
(-3+2
ϵ
8
13
3
2
ϵ
10
2
ϵ
13
2
13
3
(-1+162
ϵ
14
13
3
2
ϵ
18ϵ
13
2
3
ϵ13
6
3
ϵ13
4ϵ
13
Out[]=
Quaternion0,,,-,Quaternion0,-,,-,Quaternion0,-2ϵ,0,-,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
13-3
2
ϵ
13
3
8
13
3
2
ϵ
10
2
ϵ
13
32
3
2
ϵ
13
14
13
3
2
ϵ
2ϵ
3
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-,-,-,Quaternion,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
3
3
13
9
3
26
18ϵ
13
9
3
ϵ26
6
3
ϵ13
9ϵ
13
Out[]=
Quaternion0,-,),,Quaternion0,,-),,Quaternion0,,-,,Quaternion,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
1
13
3
(-3+2
ϵ
8
3
2
ϵ
13
10
2
ϵ
13
2
13
3
(-1+162
ϵ
14
3
2
ϵ
13
18ϵ
13
2
3
ϵ13
6
3
ϵ13
4ϵ
13
Out[]=
Quaternion0,-,,,Quaternion0,,),,Quaternion0,0,-,0,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
3
2
ϵ
13
8
3
2
ϵ
13
10
2
ϵ
13
1
26
3
(13-642
ϵ
14
3
2
ϵ
13
3
ϵ2
Out[]=
Quaternion0,,,-,Quaternion0,-,,-,Quaternion0,-2ϵ,0,-,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
13-3
2
ϵ
13
3
8
13
3
2
ϵ
10
2
ϵ
13
32
3
2
ϵ
13
14
13
3
2
ϵ
2ϵ
3
Out[]=
Quaternion0,-,,,Quaternion0,,),,Quaternion0,0,-,0,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
3
2
ϵ
13
8
3
2
ϵ
13
10
2
ϵ
13
1
26
3
(13-642
ϵ
14
3
2
ϵ
13
3
ϵ2
Out[]=
1
3
Out[]=
3
2
Out[]=
Quaternion0,,(13-3),-,Quaternion0,-,,-,Quaternion0,-2
24
3
2
ϵ
13
1
13
2
ϵ
24
2
ϵ
13
10
13
3
2
ϵ
96
2
ϵ
13
42
2
ϵ
13
3
ϵ,0,-2ϵ,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]Out[]=
Quaternion0,-,,,Quaternion0,,(13-64),,Quaternion[0,0,-ϵ,0],Quaternion,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
16
13
3
2
ϵ
2
2
ϵ
13
16
2
ϵ
13
20
2
ϵ
13
3
1
13
2
ϵ
28
2
ϵ
13
2ϵ
3
Out[]=
1
3
Out[]=
3
2
Out[]=
Quaternion0,,(13-3),-,Quaternion0,-,,-,Quaternion0,-2
24
3
2
ϵ
13
1
13
2
ϵ
24
2
ϵ
13
10
13
3
2
ϵ
96
2
ϵ
13
42
2
ϵ
13
3
ϵ,0,-2ϵ,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]Out[]=
Quaternion0,-,,,Quaternion0,,(13-64),,Quaternion[0,0,-ϵ,0],Quaternion,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
16
13
3
2
ϵ
2
2
ϵ
13
16
2
ϵ
13
20
2
ϵ
13
3
1
13
2
ϵ
28
2
ϵ
13
2ϵ
3
Out[]=
1.0472
Out[]=
2.63392
Out[]=
{Quaternion[-1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
Out[]=
{Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
Out[]=
{{Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}}
Out[]=
2.
Out[]=
0.866025
Out[]=
Quaternion1,,,-,Quaternion0,-,,-,Quaternion0,-2ϵ,0,-,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
13-3
2
ϵ
13
3
8
13
3
2
ϵ
10
2
ϵ
13
32
3
2
ϵ
13
14
13
3
2
ϵ
2ϵ
3
Out[]=
Quaternion1,-,,,Quaternion0,,),,Quaternion0,0,-,0,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
2
ϵ
13
3
2
ϵ
13
8
3
2
ϵ
13
10
2
ϵ
13
1
26
3
(13-642
ϵ
14
3
2
ϵ
13
3
ϵ2
Out[]=
{Quaternion[1.73205,0.,1.,0.],Quaternion[-0.866025,0.,1.5,0.],Quaternion[0.866025ϵ,-3.4641ϵ,-1.5ϵ,-2.ϵ],Quaternion[1.73205ϵ,-1.73205ϵ,ϵ,3.ϵ],Quaternion[0.,0.,0.,0.],Quaternion[0.,0.,0.,0.],Quaternion[0.,0.,0.,0.],Quaternion[0.,0.,0.,0.]}
Out[]=
{Quaternion[1.73205(1.+0.473373+0.284024(13.-64.)-0.00591716(13.-3.)),1.73205(-7.81065-0.12426(13.-64.)+0.0473373(13.-3.)),1.73205(0.133235+0.0444116(13.-3.)),1.73205(-1.73205+3.52559+0.0512441(13.-64.)+0.0819906(13.-3.))],Quaternion[1.73205(0.568047-0.00295858(13.-64.)(13.-3.)),1.73205(-2.+4.7929+0.0710059(13.-64.)+0.0828402(13.-3.)),1.73205(4.26351+0.0666173(13.-64.)),1.73205(-1.1547+7.76861+0.122986(13.-64.)-0.0341627(13.-3.))],Quaternion[1.73205(-2.46154+0.0384615ϵ(13.-3.)),-3.4641ϵ,-1.5ϵ,-2.ϵ],Quaternion[1.73205ϵ,1.73205(0.230769-0.0769231ϵ(13.-64.)),1.73205(-2.84234+0.0444116ϵ(13.-3.)),1.73205(-0.399704+0.133235ϵ(13.-64.))],Quaternion[0.,0.,0.,0.],Quaternion[0.,0.,0.,0.],Quaternion[0.,0.,0.,0.],Quaternion[0.,0.,0.,0.]}
4
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
4
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
4
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
4
ϵ
2
ϵ
2
ϵ
2
ϵ
4
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
4
ϵ
2
ϵ
2
ϵ
2
ϵ
2
ϵ
3
ϵ
2
ϵ
3
ϵ
2
ϵ
3
ϵ
2
ϵ
3
ϵ
2
ϵ
Out[]=
{Quaternion[6.26203-30.6337,0,0,0],Quaternion[4.46336,0,0,0],Quaternion[-4.46336,0,0,0],Quaternion[0,8.92672,-5.15385,-15.4615],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
2
ϵ
4
ϵ
2
ϵ
3
ϵ
3
ϵ
3
ϵ
3
ϵ
Out[]=
{Quaternion[0.,0,0,0],Quaternion[0.,0,0,0],Quaternion[0.,0,0,0],Quaternion[0,0.,0.,0.],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
In[]:=
(*--------------------------------------------------------------*)(*5.2NUMERICALEXAMPLE:EQUATION(5.12))*)(*DeterminationofB,M,N(=NN)*)w1=Simplify[csum[X1,X2]](*Verificationw1=B=P1*)P1w=Chop[Simplify[cdif[w1,P1]]](*ExtractionofM,N(=NN)viaB=M+ϵN*)BM=Quaternion0,0,,0,Quaternion0,0,,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]NN=Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion0,-2,-,-,Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
3
2
2
3
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-2ϵ,-,-,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
3
ϵ2
2ϵ
3
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-2ϵ,-,-,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
3
ϵ2
2ϵ
3
Out[]=
{Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion0,-2ϵ,-,-,Quaternion[ϵ,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
3
ϵ2
2ϵ
3
Out[]=
Quaternion0,0,,0,Quaternion0,0,,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
1
3
3
2
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion0,-2,-,-,Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
3
2
2
3
In[]:=
(*-------------------------------------------------------------*)(*5.2NUMERICALEXAMPLE:EQUATION(5.13))*)(*DeterminationofB,M,N(=NN),P^2=PP=-61/12*)P=Chop[Simplify[HP[miI,NN]]]PP=Chop[Simplify[HP[P,P]]]
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[1,0,0,0],Quaternion0,2,,
3
2
2
3
Out[]=
Quaternion-,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
61
12
In[]:=
(*-------------------------------------------------------------*)(*5.2NUMERICALEXAMPLE:EQUATIONS:(5.14-15)*)(*ObtentionofW1,W1.W1*)W1=Chop[Simplify[int[P,M]]]W1=Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion0,-1,,,0,0,0W1intW1=Chop[Simplify[HP[W1,W1]]]
1
3
3
,Quaternion-1
2
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion0,-1,,,0,0,0
1
3
3
,Quaternion-1
2
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion0,-1,,,0,0,0
1
3
3
,Quaternion-1
2
Out[]=
Quaternion-,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
49
12
In[]:=
(*-------------------------------------------------------*)(*5.2NUMERICALEXAMPLE:EQUATION:(5.16)*)(*ObtentionofM⊥=Mperp,M⊥^2*)WWW=Chop[Simplify[HP[M,M]]]W2=Chop[Simplify[ext[P,M]]]Pinv=Simplify[fclif[-12/61,P]]Pinv=Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion-,0,0,0,Quaternion0,-,-,-WW=Chop[Simplify[HP[P,Pinv]]]WW=Chop[Simplify[HP[Pinv,P]]]WW=Chop[Simplify[HP[Pinv,W1]]]Mperp=Quaternion0,,-,-,Quaternion0,-,,-,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]WW=Chop[Simplify[HP[Mperp,Mperp]]]
12
61
24
61
6
3
61
8
3
61
24
61
3
61
8
3
61
10
61
32
3
61
14
3
61
Out[]=
Quaternion,0,0,0,Quaternion[-1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
5
12
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion-,0,0,0,Quaternion0,-,-,
3
4
2
3
3
2
2
3
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion-,0,0,0,Quaternion0,-,-,-
12
61
24
61
6
3
61
8
3
61
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion-,0,0,0,Quaternion0,-,-,-
12
61
24
61
6
3
61
8
3
61
Out[]=
{Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
Out[]=
{Quaternion[1,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]}
Out[]=
Quaternion0,,-,-,Quaternion0,-,,-,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
61
3
61
8
3
61
10
61
32
3
61
14
3
61
Out[]=
Quaternion0,,-,-,Quaternion0,-,,-,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
24
61
3
61
8
3
61
10
61
32
3
61
14
3
61
Out[]=
Quaternion,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
49
61
In[]:=
(*---------------------------------------------------------*)(*5.2NUMERICALEXAMPLE:EQUATION:(5.17)*)(*ObtentionofW=I*W1*)W=Chop[Simplify[HP[e0123,W1]]]W=Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion,0,0,0,Quaternion0,-1,,
1
2
1
3
3
WW2=Chop[Simplify[HP[W,W]]](*----------------END------------------*)Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion,0,0,0,Quaternion0,-1,,
1
2
1
3
3
Out[]=
Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion,0,0,0,Quaternion0,-1,,
1
2
1
3
3
Out[]=
Quaternion-,0,0,0,Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0],Quaternion[0,0,0,0]
49
12
Cite this as: Patrick R. Girard, Patrick Clarysse, Romaric Pujol, Robert Goutte, Philippe Delachartre, "DUAL_HYPERQUATERNION_POINCARE_GROUPS.nb" from the Notebook Archive (2021), https://notebookarchive.org/2021-03-4mrwcdg
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