Sect. 4.5: Cayley-Klein Parameters

2. Sect. 4.5: Cayley-Klein Parameters • 3 independent quantities are needed to specify a rigid body orientation. Most often, we choose them to be the Euler Angles:,θ,ψ. • Sometimes, it’s convenient to use variable sets which contain more than the minimum number of 3, even though these can’t be used as indep generalized coords. • One set of 4 parameters, due to Klein (& Cayley) & originally Euler  Cayley-Klein Parameters is often convenient. • These are very useful in theoretical physics. • Also much easier to deal with than Euler Angles when obtaining numerical solutions to rigid body problems!

3. *  complex conjugate • Cayley-Klein Parameters: 4 complex variables denoted as: α,β,γ,δ. Constraints: β = - γ*, δ= α* • DEFINEgeneral transformation matrix A in terms of these as: • Of course, this is the SAMEA that we wrote in terms of Euler Angles!  There MUST be a connection between α,β,γ,δ& angles ,θ,ψ. • A in this form looks complex. But A is real! • PHYSICAL INTERPRETATIONofα,β,γ,δ?

4. To see that A is real, define the real quantities e0, e1, e2, e3 (Euler parameters) as: α e0 +i e3β e2 +i e1 Also, since β = - γ* & δ= α* γ -e2 +i e1 δ e0 - i e3 • In terms of these real parameters, A looks like: • Clearly, A in this form is real. • Orthogonality of aij  (e0)2 + (e1)2 + (e2)2 + (e3)2 = 1 • PHYSICAL INTERPRETATIONof e0, e1, e2, e3?

5. Sect. 4.6: Euler’s Theorem • Now have the complete math formalism to describe the motion of any rigid body. • At any time t, the body orientation with respect to some external set of axes can be specified by an orthogonal transformation A. • Can express elements of A (aij) with a convenient choice of parameters: Euler angles:,θ,ψ. Cayley-Klein Parameters:α,β,γ,δ. Euler Parameters:e0,e1,e2,e3. Each is time dependent.  As time progresses, the orientation of the body changes:A = A(t) Time dependence is obtained by solving Lagrange’s Eqtns!

6. Assume initial conditions so that body axes are the same as the external axes at t = 0. Initial condition: A(0) = 1 • Later as orientation changes, A = A(t)  1 • Physics: Motion must be continuous  The transformation matrix A must evolve continuously from the identity transformation 1.  Euler’s Theorem:The general displacement of a rigid body with one point fixed is a rotation about some axis.Proof: As we proceed.

7. Euler’s Theorem:The general displacement of a rigid body with one point fixed is a rotation about some axis. • Physical meaning:For every such rotation it is always possible to find an axis through the fixed point, oriented at particular polar angles θ,such that a rotation about this axis by a particular angle ψ duplicates the rotation.  3 parametersθ,,ψcharacterize the rotation & theseAREthe Euler angles • The fixed point is often (but not necessarily!) the CM of the body. • Take the fixed point as origin of body axes “Displacement” of body involves no translation of body axes, but only a change in ORIENTATION

8. Euler’s Theorem restated: The body axes at time t can be obtained by a SINGLE rotation (about an appropriate axis, to be determined!) of the initial axes. • In other words: TheOPERATION implied by the general orthogonal transformation A describing the motion of a rigid body IS A ROTATION! • Given the Euler angles θ,,ψ, the rotation axis is to be determined! • 1st Characteristic of a rotation: The magnitudes of all vectors are unchanged on rotation! • This results automatically from the orthogonality conditions on the aij!

9. 2nd Characteristic of a rotation: The direction of the rotation axis is unchanged on rotation. Any vector lying on this axis has the same components in both the initial & the final axis systems!  If we can show that there exists a vector R having the same components in both systems, will have proven Euler’s Theorem. This proof follows: • In general, for vector R, under a rotation characterized by A: R = AR • If R = rotation axis, then R = R • For generality, write R = λR. To prove Euler’s theorem, look for solutions where λ = +1 • Combining gives: (A - λ1)R = 0

10. To prove Euler’s theorem, we need to solve: (A - λ1)R = 0 (1)  The Eigenvalue Problem Values of λ which satisfy (1)  Eigenvalues In general, λ might be real or complex Vectors R which satisfy (1)  Eigenvectors Eigenvalue  German for characteristic value. • Euler’s theorem restated:The real, orthogonal matrix specifying the physical motion of a rigid body with one fixed point always has the eigenvalue λ = + 1

11. Solve the eigenvalue problem: (A - λ1)R = 0 (1) • Note that: X a11 a12 a13 1 0 0 R  Y A  a21 a22 a23 1  0 1 0 Z a31 a32 a33 0 0 1  (1) becomes 3 simultaneous, homogeneous, linear algebraic eqtns for the components X, Y, Z (a11 -λ)X + a12Y + a13Z = 0 a21X + (a22 -λ)Y + a23Z = 0 (1) a31X + a32Y + (a33 -λ)Z = 0 Solutions to (1): (in general 3) eigenvalues λ. For each λ, ratios of components of corresponding eigenvector R. Physics: Only the direction of R, not the magnitude, can be determined.

12. Eigenvalue problem: (A - λ1)R = 0 (1) Or:(a11 -λ)X + a12Y + a13Z = 0 a21X + (a22 -λ)Y + a23Z = 0 (1) a31X + a32Y + (a33 -λ)Z = 0 Has a solution only when the determinant of the coefficients of X,Y, & Z vanishes.  Solution requires: |A - λ1| = 0 or: (a11 -λ) a12 a13 a21 (a22 -λ) a23 = 0 (2) a31 a32 (a33 -λ) (2)  Characteristic or secular eqtn of matrix A. Euler’s theorem restated again: For real, orthogonal matrices A, the secular eqtn must have the root λ = +1

13. Eigenvalue problem, (slightly)alternate formulation: (A - λ1)R = 0 (1)  Solution requires: |A - λ1| =0 (2) • Notation: 3 eigenvalues  λk(k =1,2,3) 3 eigenvectors R  Xk (k =1,2,3) Each eigenvector Xk has 3 components labeled as Xik (Change of notation from X,Y,Z!) 1st subscript (i) labels the component 2nd subscript (k) labels the eigenvector to which the component belongs.  Eqtn resulting from (1) for kth eigenvalue (summation convention not used!): ∑jaijXjk = λkXik (3)

14. Eigenvalue problem:∑jaijXjk = λkXik (3) • Rewrite using δj,knotation:  ∑jaijXjk = ∑jXij δj,kλk (3) (3): Both sides have the form of matrix products: Define the (diagonal) eigenvalue matrix: λ1 0 0 λ 0 λ2 0 (3) becomes: AX = Xλ (4) 00 λ3 • Multiply (4) from left by X-1: X-1AX = λ (5) (5): A similarity transformation operating on A.  Can diagonalize A by performing a suitable similarity transformation. If an appropriate X can be found, the elements of diagonal A  are the eigenvalues sought & the X’s which do this are the eigenvectors.

15. A proof of Euler’s Theorem in form: “For real, orthogonal matrices A, the secular eqtn must have the root λ = +1” • Diagonalize A & find eigenvalue λ = +1. • Another proof: Use property of transpose Ã. Recall for orthogonal matrices, the reciprocal is equal to the transpose: A-1 = Ã • Consider the expression (A - 1)Ã = 1 - Ã. • Take the determinant of both sides: |A - 1||Ã| = |1 - Ã| • To describe rigid body motion, A(t) must correspond to a proper rotation |A| = |Ã| =1 |A - 1| = |1 - Ã|. Determinant of a matrix = determinant of its transpose:  |A - 1| = |1 - A|  Determinant of matrix A - 1 = determinant of matrix 1 - A = -(A - 1)

16. |A - 1| = |1 - A| = -|A -1| |A -1| = 0. Compare to secular eqtn |A - λ1| = 0 A must always have at least one eigenvalue λ = + 1. Euler’s Theorem is proven! • What about the other 2 eigenvalues? • Determinant of any matrix is unaffected by similarity transformation. We had: AX = Xλ (4) andX-1AX = λ (5) Take determinant of (5), noting that |A| = 1 (previous result) & that determinant is invariant under similarity transformation: |A| = 1 = |λ| = λ1λ2λ3

17. Determinant of A = product of its 3 eigenvalues: |A| = 1 = λ1λ2λ3 • Euler’s theorem: At least one eigenvalue is 1 (say λ3 = 1) λ1λ2 = 1 • A is real  If λ is an eigenvalue, then it’s complex conjugate λ* is also an eigenvalue.  If the eigenvalue λ1 is complex, must have λ2 = λ1*. • If the eigenvalue λ1 is complex, the corresponding eigenvector R1is also complex. • For complex vectors R, the square of the magnitude is given by RR*

18. The square of the magnitude is invariant under a real orthogonal transformation A  RR* = (AR)AR* = RAAR* = RR* (1) • If R is a complex eigenvector corresponding to a complex eigenvalue λ, the first part of (1) becomes: RR* = λλ* RR* (2) (1) & (2) together  λλ* =1 Conclusion: All eigenvalues of a general orthogonal transformation A have unit magnitude.

19. Summary: The 3 eigenvalues λ1, λ2, λ3of an orthogonal transformation matrix A must satisfy: a.λ1λ2λ3 = 1 b.Euler’s Theorem One of them (say λ3) = 1 c.λ1λ2 = 1 d.λiλi* =1(i =1,2,3)  Theλ’shave 3 possible distributions: 1.All are = +1.  A = 1 (trivial) 2. One, say λ3= 1 & the other 2, λ1=λ2= -1  A = rotation by π about some axis 3.One, say λ3= 1 & the other 2, λ1&λ2are complex conjugates of each other.

20. Consider case 3. where λ3= 1 & λ2= λ1*: Still must have λ1λ2 = 1  λ1 &λ2must be of the form λ1= eiΦλ2= e-iΦ • Direction cosines of axis of rotation (eigenvector R for eigenvalue λ= 1) are obtained by going back to eigenvalue eqtns: (a11 -λ)X + a12Y + a13Z = 0 a21X + (a22 -λ)Y + a23Z = 0 (1) a31X + a32Y + (a33 -λ)Z = 0 setting λ= 1 & solving for X, Y, Z.

21. Can also get angle of rotation for eigenvalue λ= 1. Do this by a similarity transformationBAB-1  A to transform A into a system of coords where z axis is axis of rotation. A  a rotation about the z axis through angle Φ Can easily write: cosΦ sinΦ 0 A = -sinΦcosΦ 0 0 0 1 Trace of A = TrA = aii = 1 + 2 cosΦ • The trace of a matrix is invariant under a similarity transformation TrA = aii = 1 + 2 cosΦ

22. TrA = aii = 1 + 2 cosΦ • Again using the same property, this TrA = Trλ = 1 + 2 cosΦ (1) • Now, the rotation angleΦ can clearly be seen to be identical to the phase angle of the complex eigenvalues: λ1= eiΦ, λ2= e-iΦ , λ3= 1 TrA = Trλ = 1 + eiΦ + e-iΦ (2) (1) & (2) are the same, since eiΦ + e-iΦ 2 cosΦ • Clearly, cases when eigenvalues are all real, are special cases of complex eigenvalues (special choices for rotation angle Φ). Φ = 0  All λ’s =1  A = 1 (as already noted) Φ = π λ1= λ2= -1 (as already noted)

23. Note: The prescription for getting the rotation axis direction R & for rotation angle Φ are not unique & unambiguous. • If R is eigenvector, so is -R  Sense of direction of rotation axis is not exactly specified. Could be direction of R or -R • Also, if replace Φby -Φ all of the formalism is unchanged. (e.g. -Φ satisfies TrA = 1 + 2 cosΦ)  The sense of direction of the rotation angle is not exactly specified. Could be Φor - Φ. • Can go even further & say that the eigenvalue eqtn does not uniquely specify the orthogonal transformation matrixA . e.g. can show that the inverse or transpose A-1 = Ã has the same eigenvalues & eigenvectors as A.

24. Euler’s Theorem:The general displacement of a rigid body with one point fixed is a rotation about some axis. • Corollary  Chasles’ Theorem:The most general displacement of a rigid body is a rotation about some axis plus a translation. • Removing the constraint of the one fixed point introduces 3 more (translational) degrees of freedom (3 more generalized coords, giving a total of 6, as discussed at beginning of the chapter). • A stronger form (footnote, p 161): It is always possible to choose the origin of the body set of coordinates so that the translational motion is in the same direction as the rotation axis  Screw motion. Useful in crystallography!