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:

aij cosθij: a11 a12 a13

A a21 a22 a23

a31 a32 a33

• 9 aij , 6 orthogonality relns: aijaik = δj,k

3 indep functions of aij could be chosen as indep generalized coords. Choice of these is ~ arbitrary. Here, we discuss conventional choice: Euler Angles

• We found, for determinant of general orthogonal transformation A : |A| = 1

• In addition to orthogonality relns: aijaik = δj,k

another requirement that matrix elements aij 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

• 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 -1 Clearly, |S| = -1

0 0 -1• Applying S to coords gives x = Sx

xi = - xj (i = 1,23)

S Inversion transformation

Changes sign of each component and changes right hand coord system into a left handed one.

• 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 1 0 0 -1 0 0 -1

• 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

• 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.

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.

• 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

• Picture of all 3 rotations: Euler Angles ,θ,ψ

• 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

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

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”

Rotation #3. Rotate (ξ,η,ζ) by angle ψ about the ζ axis. New coords (x,y,z) See figure. Orthogonal transformation B

x BξEasy to show:

cosψ sinψ 0

B = -sinψ cosψ 0

0 0 1

Total: Three successive rotations.

x Ax

with A BCD

Can show

(student exercise!):

Total: x Ax

with

Of course, the inverse transformation is:

x A-1x . Since, by earlier discussion

A-1 = Ã Transpose of A, can get:

• Note: the sequence of rotations is ~ arbitrary.– Initial rotation: Could be taken about any of the 3 Cartesian

axes.

– 2nd & 3rd 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 2nd 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”

– 3rd 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!