Topic 1: Quaternions Shadows of Shadows

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1 Lectue Seies: The Spi of the Matte, Physics 450, Fall 00 Topic : Quateios Shadows of Shadows D. Bill Pezzaglia CSUEB Physics Updated Nov 8, 00 Idex A. Hamilto s Quateios B. Reflectios C. Rotatios D. Refeeces

2 A. Hamilto s Quateios 3. Two i s ae bette tha oe. Quateioic Algeba 3. ecto Poducts. Two i s ae bette tha oe 4 (a) Agad Diagam Complex Algeba ca epeset D vecto Hamilto poposed a secod imagiay j fo the 3 d dimesio i = j =

3 b. The Discovey 5 Evey moig i the ealy pat of the abovecited moth, o my comig dow to beakfast, you (the) little bothe William Edwi, ad youself, used to ask me: "Well, Papa, ca you multiply tiplets?" Wheeto I was always obliged to eply, with a sad shake of the head: `No, I ca oly add ad subtact them". Si William Rowa Hamilto Octobe 6, 843, while walkig with his wife alog the Royal Caal to a meetig of the Royal Iish Academy i Dubli, the solutio came to him ad he caved it ito the stoe of the Bougham Bidge. c. Aticommutivity is Pepediculaity 6 No-commutative multiplicatio i j = j i Example: coside a D vecto = 3i + 4j The squae of a D vecto has a Pythagoea om! = 3 i +4 j + 3 4( i j + j i ) = -(3 +4 )= -5 3

4 c. The Thid i 7 The poduct of the two imagiaies is somethig ew k = ij = ji It follows that it too is imagiay k = ( ij)( ij) = ijji = Also, it aticommutes with the othes, so they ae mutually pepedicula! ki = ik = j jk = kj = i The famous equatio witte at bidge 8 4

5 . Quateioic Algeba 9 (a) A quateioic umbe Has 4 pieces Hamilto segegates ito scala ad vecto pats (b) Hemetia Cojugate 0 Aalogous to complex cojugate i = -I j = -j k = -k Fo this to wok, the cojugate must be a ati-semiliea ivolutio (ab) = b a 5

6 (c) Decompositio of Quateio Give Quateio: H=a +bi +cj +dk Cojugate is H =a bi cj dk Scala Pat: ½(H+H ) = a ecto Pat: ½(H-H ) = bi +cj +dk (d) Noms ad Iveses Give Quateio: H=a +bi +cj +dk Modulus: H =HH =a +b +c +d Ivese is hece: H H = HH + + The om shows that quateios ca epeset a 4D Euclidea space. Havig iveses, quateios ae a divisio algeba 6

7 3. ecto Algeba (a) Geometic (diect) Poduct 3 Coside two vectos A= a i +a j B= b i +b j Multiply them AB =(a i +a j)(b i +b j) = (a b +a b ) + (a b -a b )k O, i tems of Gibb s vecto otatio: AB = A B + A B (b) Defiitio of Poducts Poduct Revese AB = A B + A B BA = A B A B Dot poduct is symmetic itechage A B = + B A = 4 ( AB + BA) = A, } { B Coss poduct is atisymmetic itechage A B = B A = ( AB BA) = [ A, ] B 7

8 (c) Tiple Poducts You ca show the scala pat of the poduct of thee vectos is the volume of the paalleliped (i.e. the tiple scala poduct) ( ABC CBA) = A ( B ) C 5 Fom commutato idetities you ca easily deive the BAC-CAB ule A ( B C) = [ A,[ B, C]] 4 ({ B,{ A, C}} { C,{ A, B}} ) = 4 = B( A C) C( A B) B. Reflectios 6. Geometic Defiitio. Quateioic Method 3. Two Reflectios make a Rotatio 8

9 . Geometic Defiitio 7 (a) Reflectio aoud Y axis Is equivalet to otatig 80 aoud the y axis (y axis is fixed ) x x 0 y = + y z z (b) Reflectio about abitay axis 8 Decompose vecto ito pats paallel ad pepedicula to eflectio axis The eflected vecto is: ' ˆ = = + 9

10 0 (c) Fomulatio Costuct the pieces The eflected vecto is hece I matix fom, if axis vecto is: 9 ˆ ˆ) ( = = = ' = ' ' ' z y x z y x ),, ( ˆ 3 =. Quateio Reflectios (a) Reflectio uit axis: (b) Reflectio Fomula is simple: 0 k j i 3 ˆ + + = ˆ = ˆ ˆ ' = Fom this it is easy to pove that eflectio of poduct is poduct of eflectio!

11 3. A Rotatio is two eflectios (a) Two eflectios make a otatio A compositio of eflectios ove lies itesectig with agle θ is equivalet to a otatio of agle θ about the axis which is pepedicula to the plae descibed by the lies Who did this fist? Was it Eule? (b) Cayley s Fom of Rotatios Let a ad b be uit (quateio) vectos which descibe plae of otatio, with agle betwee them: θ/ Compositio of eflectios of vecto is simply: =baab

12 C. Rotatios 3. How to Rotate. Compositio of Rotatios 3. The eds do ot dictate the meas. How to Rotate 4 (a) D Rotatio 793 Wessel shows multiplicatio by i otates complex vecto by 90 Rotate by agle φ: I Catesia Fom: z' = e iφ ( x ' + iy') = (cosφ + i siφ)( x + iy) z

13 Matix Repesetatio Athu Cayley defies matix multiplicatio algeba Matix fom of otatio about z axis by agle φ: x' cosφ y' = siφ z' 0 siφ cosφ 0 0 x 0 y z (b) Thee Dimesioal Rotatio 6 775: Eule s otatio theoem: ay geeal otatio (oe fixed poit) ca be descibed by a sigle otatio about a fixed axis (the Eule Pole ). Hece thee ae 3 paametes, two descibig the diectio of the axis ad the thid beig the magitude of the otatio about that axis. Leohad Eule

14 Fomula fo Geeal 3D Rotatio 7 Eule s wok was befoe vectos. J. Willad Gibbs Gibbs (88?) expessed the geeal coical otatio about axis i vecto fom: ' = cosφ + ˆ( ˆ )[ cosφ] + ( ˆ) siφ Matix fo Geeal 3D Rotatio 8 Let otatio descibed by axis vecto: φ = φˆ = ( φ, φ, φ3 ) The matix fom of otatio is quite ugly aij = si φ φ δ ij cos φ + ε ijk φk + cos φ φiφ j φ 4

15 (c) Quateioic Repesetatio Cayley shows double eflectio otatio fomula is equivalet to biliea tasfomatio with quateio q ' = ( ba ˆˆ) ( ab ˆ ˆ) = qq Quateio q is elated to the two eflectio axes: q = aˆ bˆ + aˆ bˆ φ = cos ˆ + si Thus otatio about axis by agle φ has pleasig fom: φ = e ˆ φ / ' = e e ˆ φ / ˆ φ /. Compositio of Rotatios (a) 775: Eule shows that two successive otatios about diffeet axes ca be expessed as a sigle otatio about a fixed axis. Example: two eflectios (i.e. 80 otatios) make a otatio Leohad Eule Geeal compositio of otatios is a eal mess 5

16 (b) Quateioic Compositio 3 Cayley shows that the poduct of two quateios is a quateio which epesets the compositio of otatios! ' = R( b) R( a) R ( a) R ( b) Rotatios: Net Rotatio R ( a ) = e R( c) ˆ aα / cˆ / = e γ R ( b) = e = R( b) R( a) ˆ bβ / (c) Rodigues Half-Agle Fomula 3 Explicitly multiplyig the quateios gives you a way to detemie axis c ad agle γ α α β β γ γ (cos + a ˆsi )(cos + bˆsi ) = (cos + cˆsi ) 840, thee yeas befoe Hamilto s quateios, Rodigues got this esult geometically (the Eule-Rodigues paametes ae essetially the compoets of quateios) cˆ ta γ = aˆ ta α ˆ β + b ta + aˆ bˆ ta aˆ bˆ ta ta α β α Olide Rodigues ta β 6

17 3. The eds do ot dictate the meas 33 (a) The ode of taslatios does ot matte. 3 ove the 4 up 4 up, the 3 ove Move 5 at agle 53 Hece o matte what path you took, you would assig coodiates (3,4) to the edpoit Equivaletly, statig at (0,0), as log as you evetually cak to (3,4) you get to same place (b) Ode of otatios DOES matte 34 If you evese ode of otatios you ed up i a diffeet oietatio. R( b) R( a) R( a) R( b) Hece the coodiates you would assig to a paticula edig place deped upo the path, the histoy of how the otatios wee doe. 7

18 (c) Eule Agles 35 By doig thee otatios i a specific ode, Eule ca assig a uique set of coodiates to a paticula oietatio. I quateio fom the opeato is quite simple: R ( φ, θ, ψ ) = e kˆ ψ / e iˆ θ / e kˆ φ / Matix fom of Eule Agles 36 I matix fom, the geealized Eule otatio is quite ugly. cosψ cosφ cosθ siφ siψ siψ cosφ cosθ siφ siψ siφ siθ cosψ siφ + cosθ cosφ siψ siψ siφ + cosθ cosφ cosψ siθ cosφ siψ siθ cosψ siθ cosθ 8

19 Summay: Why use Quateios? 37 Eule agles suffe fom gimbal lock Itepolatig betwee two states with Eule agles gives uphysical esults, wheeas Quateios look ight. Compositio of otatios is less complex (faste) with quateios Eule matix multiplicatio suffes fom tucatio eos that cause dift eos, ad size eos. Appedix 38 Eule s Theoem: Whe a sphee is moved aoud its cete it is always possible to fid a diamete whose diectio i the displaced positio is the same as i the iitial positio. 9

20 Refeeces 39 Tou of the Boombidge: Code fo Quateios ad Rotatios: Itesectio Lies Theoem: P. Nahi, Olive Heaviside (Joh Hopkis Uiv Pess, 00), see chapte 9 fo a good summay of The Geat Quateioic Wa (the debate of quateios vs vectos). Notes 40 I Maxwell s Teatise o Electicity & Magetism, vol, p6, he says that goig fom left haded to ight haded system, i.e. a eflectio, is a pevesio (attibutes to Listig, ad otes Hamilto used a left haded system. 0