2. Sect. 4.7: Finite Rotations
So far: Have used various representations to describe the relative orientation of 2 Cartesian coord systems with common origin: We’ve shown the transformation matrix A in terms of:
The Euler Angles: A = A(,θ,ψ)
The Cayley-Klein parameters: A = A(α,β,γ,δ)
Euler parameters: A = A(e0,e1,e2,e3)
Single rotation about a given direction A = A(Φ)
Here: We seek another representation: A vector relationship between an initial vector r & the vector r which results from the orthogonal transformation (rotation) r = Ar. Representation in terms of the rotation angle Φ & direction cosines of rotation axis.

3. Get result using figure &
vector algebra. Treat
transformation as active
 Counterclockwise coord
rotation = Clockwise
rotation of vectors in the
fixed coord system.
Fig: Initial position of vector r = OP. Final position
r = OQ. Rotation axis along ON. Unit vector along ON
= n. Rotation angle = Φ. |ON| = nr  ON = n(nr).
NP = OP - ON = r - n(nr). |NP| = |NQ| = |rn|
OQ = r = ON + NV + VQ

4. Taking all of these together:
r = n(nr) + [r - n(nr)]cosΦ + (rn)sinΦ Or:
r = r cosΦ + n(nr)(1 - cosΦ) + (rn)sinΦ
 “The Rotation Formula”
Holds for any rotation!

5. Worth mentioning: Can easily express rotation angle Φ in terms of Euler angles ,θ,ψ by comparing representations of the transformation matrix A in terms of one & the other.
For A = A(,θ,ψ) we had:
For A = A(Φ) we had:
cosΦ sinΦ 0
A= -sinΦ cosΦ 0

6. The trace of A is invariant under a change of representation:
 TrA(Φ) = TrA(,θ,ψ)
Use some trig identities (student exercise!):
 cos[(½)Φ)] = cos[(½)( + ψ)]cos[(½)θ]
Sign fixed by requiring Φ  0
as ,θ,ψ  0

7. Sect. 4.8: Infinitesimal Rotations
Summary: Representations to describe relative orientation of 2 Cartesian coord systems with common origin: r = Ar
Euler angles: A = A(,θ,ψ)
Cayley-Klein parameters: A = A(α,β,γ,δ)
Euler parameters: A = A(e0,e1,e2,e3)
Single rotation about a given direction A = A(Φ)
Direct vector relationship between r & r
r = r cosΦ + n(nr)(1 - cosΦ) + (rn)sinΦ
In all cases discussed for the transformation A, the number of matrix elements is > # independent variables (=3)  Had a number constraints between variables to reduce # indep ones to 3.

8. In proof of Euler’s Theorem we’ve shown: Any given orientation r = Ar can be obtained by a single rotation about some axis.
 It would be nice if we could associate a vector (with 3 independent components) with a finite rotation of a rigid body about a fixed point.
Direction of vector is obvious: Direction of axis of rotation. That is, direction of eigenvector R.
Magnitude: A suitably chosen function of rotation angle Φ could possibly be used as magnitude of this vector.
However, we cannot do this (for finite rotations!).

9. Vectors  Addition must be commutative: “A” + “B” = “B” + “A”
Why?
Let “A” & “B” be 2 vectors associated with transformations A & B.
Vectors  Addition must be commutative: “A” + “B” = “B” + “A”
But, we’ve seen, the “addition” of 2 rotations (one, followed by a 2nd) corresponds to product AB
Matrix multiplication is NOT commutative:
AB  BA
 A & B are not commutative & cannot be represented as vectors.

10. That is, the “sum” of finite rotations depends on the order of rotations. See figure for illustration:

11.  A finite rotation cannot be represented by a vector.
However, we can show that infinitesimal rotations CAN be represented by a vector!
Infinitesimal rotation  An orthogonal transformation of coordinate axes x = Ax in which components of the vector are almost the same with respect to both sets of axes (infinitesimal change).
That is, xi  xi + small corrections (i = 1,2,3)
Write: xi = xi + εi1x1 + εi2x2 + εi3x (i = 1,2,3)
Where the εij are infinitesimals.  In calculations only terms linear in εij are kept, higher powers are neglected.

12. Using summation convention, write infinitesimal transformation as: xi = (δij+ εij)xj (i = 1,2,3)
In matrix notation (defining infinitesimal matrix ε & relating it to transformation matrix A) this is:
x  (1+ ε)x  Ax
That is: (1+ ε)  A
Note that the sequence of operations doesn’t matter for infinitesimal transformations ε1 + ε2 (they commute because powers of ε higher than 1st are neglected):
(1+ ε1)(1+ ε2) = ε11 + ε21 + ε1ε21
Or: (1+ ε1) (1+ ε2)  1 + ε1 + ε2
 The order of infinitesimal rotations doesn’t matter!

13. Infinitesimal rotations: (1+ ε)  A
In terms of Euler angles ,θ,ψ, orthogonal matrix A is represented as:
For infinitesimal Euler angles: d,dθ,dψ:
cos(d)  cos(dθ)  cos(dψ)  1
sin(d)  d, sin(dθ)  dθ, sin(dψ)  dψ
 For infinitesimal rotations:
(d + dψ) 0
A = - (d + dψ) dθ dθ

14. Infinitesimal rotations: (1+ ε)  A, ε = A - 1 Where: 1 (d + dψ) 0
A = - (d + dψ) dθ dθ
(d + dψ) 0
 ε = A = - (d + dψ) dθ dθ
Euler angle representation of ε

15. General properties of ε & A: ε = A - 1, 1+ ε = A
Clearly, A-1 = Ã = 1- ε
Proof: AA-1 = (1+ ε)(1- ε) = 12 + εε  1
Also: ε = - ε. ε is antisymmetric!
Proof: Ã = 1- ε = 1 + ε  ε = - ε
Convenient to write ε in form:
0 dΩ dΩ2
ε = - dΩ dΩ1
dΩ2 - dΩ
dΩ1, dΩ2, dΩ3  3 infinitesimal parameters identified with 3 independent parameters needed to specify rotation.

16. We had: r  (1+ ε)r  Ar (or x = (1+ ε)x = Ax)  r - r = εr  dr
Using the ε matrix, this becomes:
dx1 = x2dΩ3 - x3dΩ2
dx2 = x3dΩ1 - x1dΩ3
dx3 = x1dΩ2 - x2dΩ1
These expressions should remind us of components of the cross product of 2 vectors.
 Define the vector: dΩ  vector with components dΩ1, dΩ2, dΩ3
Text emphasizes that dΩ  differential of a vector! Instead, it is a vector of differential magnitude. That is a vector Ω which has a differential dΩ does not exist!!!!

17. So, we can write: dr = εr  r  dΩ
r is a vector  Transforms under an orthogonal transformation B according to r = Br or:
xi = bijxj
If dΩ is a vector, it must also transform like this under an orthogonal transformation B. Will now look at the transformation properties of dΩ
To look at this, first look at transformation properties of matrix ε under the transformation B. Previous discussion  ε = BεB-1
It is shown in Prob. 3, p 180, that the property that ε is antisymmetric is preserved in this similarity transformation.

18. So similar to ε, ε can also be written as:
0 dΩ3 - dΩ2
ε = -dΩ3 dΩ1
dΩ2 - dΩ1
Skipping some tedious steps, can show that dΩi are related to dΩj by (|B|  determinant of B) :
dΩi = |B|bijdΩj
This transformation is ALMOST the same as for a vector (xi = bijxj) but differs by the factor of |B|.

19. Before discussing the details of the difference between dΩ and a vector, look at its properties another way.
For FINITE rotations, we had:
r = r cosΦ + n(nr)(1 - cosΦ) + (rn)sinΦ
 “The Rotation Formula” (1)
To get the present INFINITESIMAL rotation case from this, let Φ  dΦ in (1).
 cos(dΦ)  1, sin(dΦ)  dΦ
 (1) becomes: r = r + (rn)dΦ
Or: r - r  dr = (rn)dΦ
We had: dr  r  dΩ  dΩ  ndΦ

20. dΩ = ndΦ under the orthogonal transformation B:
Now, discuss transformation properties of
dΩ = ndΦ under the orthogonal transformation B:
We had: dΩi = |B|bijdΩj (1)
Ordinary vectors transform like: xi = bijxj (2)
Define: (property under the transformation B):
Polar Vector  Any quantity for which the components transform according to (2)
Axial Vector  Any quantity for which the components transform according to (1)
(also called a pseudovector)
Properties under inversion Sij = -δij:
Polar Vector: All components change sign
Axial Vector: Components do not change sign.

21. Sect. 4.8: Polar & Axial Vectors
Example of an axial vector: A vector which is the cross product of 2 polar vectors. If D, F are polar vectors: V* = D  F is an axial vector.
Clearly, Vi* = DjFk -DkFj (i,j,k cyclic)
By definition of polar vectors, components Dj, Fk, Dk, Fj change sign on inversion  Vi* does not.
Physical quantities which are polar vectors:
Position r, velocity v, momentum p = mv,
acceleration a, force F = ma, electric field E,...
Physical quantities which are axial (pseudo) vectors: Angular momentum L = r  p, torque
N = r  F, magnetic field B, ...

22. S = scalar, V = vector, V* = axial (pseudo) vector
Define a Parity Operator P (actually ~ same as inversion operator of earlier): P converts x, y, z to -x,-y, -z
S = scalar, V = vector, V* = axial (pseudo) vector
 PS = S, PV = -V, PV* = V*
Suppose, we have a scalar (like) quantity S*  VV*
PS* = P(VV*) = -(VV*) = -S*
Pseudoscalar  Any otherwise scalar quantity which behaves like S* under inversion.
Any quantity which can be written as the scalar product of a polar vector and an axial (pseudo) vector.
Physical quantities which are pseudoscalars: Magnetic flux, ….

23. Polar vectors: The vector itself remains unchanged on inversion, the components change sign.
Axial vectors: Carry a “handedness” property with them (consistent with the cross product!). Cross product (the vector itself!) clearly changes sign with coordinate inversion.
Back to infinitesimal rotations:
We had: dΩi = |B|bijdΩj (1)
Ordinary vectors transform like: xi = bijxj (2)
Also: dr  r  dΩ. Both r & dr are polar vectors & transform according to (2).  dΩ must be an axial vector, transforming according to (1).
dΩ  ndΦ. Inversion  n changes direction.

24. Some comments on cross product. C = D  F
Introduce the Levi-Civita density ijk
Write Cartesian components of C:
Ci  ∑j,kijkDjFk
ijk  permutation symbol or Levi-Civita density
ijk  0, if any 2 indices equal
 1, if i, j, k are even permutation of 1,2,3
 -1, if i, j, k are an odd permutation of 1,2,3
122 = 313 = 211 = 0 , etc.
123 = 231 = 312 =1, 132 = 213 = 321 = -1

25. Formalism developed so far (r = Ar) to represent the orientation of a rigid body. Transformation involves rotation of the coordinate system. That is, usually assumed “passive” interpretation of transformation A. Counterclockwise rotation about some axis.
Often need “active” interpretation of r = Ar. Physical system & the associated vectors are rotating in the clockwise direction.
Often also, need rotation formulas explicitly for this but for vectors rotating in counterclockwise direction. If careful, can use formulas just derived. But, need to let direction of rotation be opposite & also axis of rotation to be opposite.

26. Figure illustrates passive, active transformations:

27. Formulas for Counterclockwise Rotations (note sign changes in Φ, dΩi)
“The Rotation Formula”
r = r cosΦ + n(nr)(1 - cosΦ) - (rn)sinΦ
Infinitesimal rotations: dr = - (rn)dΦ
Antisymmetric, infinitesimal rotation matrix ε
0 - dΩ3 dΩ2
ε = dΩ dΩ1
- dΩ dΩ
Or, using dΩ  ndΦ
0 -n3 n (ni = components of n)
ε = n n1 dΦ
n2 n  

28. Using dr = - (rn)dΦ we can immediately get the derivative of r with respect to Φ:
(dr/dΦ) = - (rn) (1)
Define the matrix N: Nr  (rn)
 (dr/dΦ) = -Nr (2)
Using the cross product definition (& summation convention):
(rn)i  ijk xjnk
 Nij = ijk nk

29. Another (sometimes) useful representation for matrix ε:
Start from n3 n2
ε = n n1 dΦ (ni = components of n)
n2 n
Define matrices Mi  Infinitesimal rotation generators
M1  , M2  , M3 
Can show: MiMj - MjMi  [Mi,Mj] = ijk Mk (cyclic i,j,k)
[Mi,Mj]  Commutator  Lie Bracket
These relations define the Lie algebra of the rotation group. Relations like these come up all the time in QM (where the M’s are matrix representations of angular momentum)!
Finally can write: ε = niMidΦ (summation convention!)