2. Sect. 4.4: Euler Angles Back to PHYSICS! GOAL: Describe rigid body motion (especially rotations) with a Lagrangian formalism. What generalized coordinates to use? General coord transformation, described by matrix A. Representation of A in terms of direction cosines of new axes with respect to old: a ij  cosθ ij : a 11 a 12 a 13 A  a 21 a 22 a 23 a 31 a 32 a 33 9 a ij, 6 orthogonality relns: a ij a ik = δ j,k  3 indep functions of a ij could be chosen as indep generalized coords. Choice of these is ~ arbitrary. Here, we discuss conventional choice: Euler Angles

3. We found, for determinant of general orthogonal transformation A : |A| =  1 In addition to orthogonality relns: a ij a ik = δ j,k another requirement that matrix elements a ij of must satisfy to describe rigid body motion: Must have |A| = +1 Mathematically, |A| = -1 is allowed. However, PHYSICALLY, cannot describe rigid body motion with an A which has |A| = -1 Proper Transformations  Orthogonal transformations with |A| = +1 Improper Transformations  Orthogonal transformations with |A| = -1

4. Discussion of why A with |A| = -1 cannot correspond to a physical rotation of a rigid body: Consider a specific transformation described by -1 0 0 S = 0 -1 0  -1Clearly, |S| = -1 0 0 -1 Applying S to coords gives x = Sx  x i = - x j (i = 1,23)  S  Inversion transformation Changes sign of each component and changes right hand coord system into a left handed one.

5. How can an inversion be achieved by a series of rotations & reflections? One way (figure): Two step process: 1. Rotate about a coord axis (say z) by 180º (x  -x, y  -y) 2. Reflect in that (z) axis direction (z  -z) Represent in terms of orthogonal transformation matrices: AB = S where -1 0 0 1 0 0 -1 0 0 A = 0 -1 0 B = 0 1 0 S = 0 -1 0 0 0 10 0 -1 0 0 -1

6. Inversion: -1 0 0 S = 0 -1 0 0 0 -1 Cannot get this by a rigid change of axes orientation!  Inversion S can never correspond to a physical displacement of a rigid body. Also true for ANY transformation A with |A| = -1. True because any A like this can be written A = A S where |A| = +1 (since |A| = |A||S| = - |A|)  Transformations A representing rigid body motion MUST have |A| = +1

7. Goal: Describe motion of rigid bodies in a Lagrangian formulation.  Seek 3 indep parameters (generalized coords) to specify rigid body orientation. Require resulting orthogonal transformation A have |A| = +1 Once such generalized coords are found, write Lagrangian in terms of them. No unique choices for these coords. Most common & useful: Euler (Eulerian) Angles.

8. Euler Angles Can express general orthogonal transformation from one coord system (x,y,z) to another (x,y,z) by three successive counterclockwise rotations, carried out in a specific sequence. Euler angles are then defined as the 3 rotation angles. –Convention for the choice of rotation sequence & of rotation angles is arbitrary. See text’s extensive discussion, p 151 & p 154, of this fact. See also footnote, p. 152. –Goldstein’s choice is a convention which is used in celestial mechanics, applied mechanics, solid state physics. For other conventions: See p. 152 & Appendix A.

9. General transformation from (x,y,z) to (x,y,z): 3 successive COUNTERCLOCKWISE rotations: 1. Rotate initial (x,y,z) by an angle  about the z axis. Call new coords (ξ,η,ζ) “(xsi,eta,zeta)”. The orthogonal transformation which does this  D. ξ  Dx 2. Rotate (ξ,η,ζ) by an angle θ about the ξ axis. Call new coords (ξ,η,ζ). The orthogonal transformation which does this  C. ξ  Cξ. The ξ axis is at intersection of the x-y and ξ- η planes  “LINE OF NODES” 3. Rotate (ξ,η,ζ) by an angle ψ about the ζ axis to produce (x,y,z) coords. The orthogonal transformation which does this  B. x  Bξ

10. Picture of all 3 rotations: Euler Angles  ,θ,ψ

11. Express each successive rotation as an orthogonal transformation matrix. Multiply to get the total transformation. ξ  Dx, ξ  Cξ = CDx x  Bξ = BCDx Or: x  Ax where A  BCD Since we already know the form of the transformation matrix for rotation through one angle, it’s easy to write down the matrices D,C,B. Then, matrix multiply to get A = BCD

12. Rotation #1. Rotate (x,y,z) by angle  about z axis. New coords  (ξ,η,ζ). See figure. Orthogonal transformation  D. ξ  Dx Easy to show: cos  sin  0 D = -sin  cos  0 0 0 1

13. Rotation #2. Rotate (ξ,η,ζ) by angle θ about ξ axis. New coords  (ξ,η,ζ). See figure. Orthogonal transformation  C ξ  Cξ Easy to show: 1 0 0 C = 0 cosθ sinθ 0 -sinθ cosθ ξ axis is at the intersection of the x-y & ξ-η planes  “LINE OF NODES”

14. Rotation #3. Rotate (ξ,η,ζ) by angle ψ about the ζ axis. New coords  (x,y,z) See figure. Orthogonal transformation  B x  Bξ Easy to show: cosψ sinψ 0 B = -sinψ cosψ 0 0 0 1

15. Total: Three successive rotations.  x  Ax with A  BCD Can show (student exercise!):

16. Total: x  Ax with Of course, the inverse transformation is: x  A -1 x. Since, by earlier discussion A -1 = Ã  Transpose of A, can get:

17. Note: the sequence of rotations is ~ arbitrary. –Initial rotation: Could be taken about any of the 3 Cartesian axes. –2 nd & 3 rd rotations: The only limitation = no 2 successive rotations can be about the same axis.  12 possible conventions to define Euler angles! (For right handed coord systems! If we include left handed coord systems, 12 more!) –The 2 most common ones: Differ in the axis choice for the 2 nd rotation (θ). Here: we rotate about intermediate x axis (ξ axis)  “x convention” In QM often take this rotation about intermediate y axis (η axis)  “y convention” –3 rd convention (engineering): Used to describe the orientation of moving air (or space) craft. See p. 154 for discussion. See also Appendix A for more conventions!