Euler angles Three giants in the theory of rotations

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Transcript of Euler angles Three giants in the theory of rotations

Vector Waves and Probe Compensation - Lecture 5: Rotations of Vector WavesRotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Assignment
Vector Waves and Probe Compensation Lecture 5: Rotations of Vector Waves
Gerhard Kristensson
Lund University
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Leonhard Euler (1707–1783), Swiss mathematician and physicist
Sir William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathemati- cian
Eugene Paul Wigner (1902– 1995), Hungarian American physicist and mathematician (Nobel Prize in Physics in 1963)
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 1/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
e1
e2
e3
e′1
e′2
e′3
A rotation is characterized by an axis of rotation and an angle of rotation (proved by Euler)
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 2/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Two coordinate systems — rotated II
The two sets of units vectors are linearly related e′1 = e1a11 + e2a12 + e3a13
e′2 = e1a21 + e2a22 + e3a23
e′3 = e1a31 + e2a32 + e3a33
or
The coefficients aij are the direction cosines
The inverse is (aij is an orthogonal matrix) [8]
ei = 3∑
j=1
e′jaji, i = 1, 2, 3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 3/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
The position vector transforms (two rectilinear systems)
r = 3∑
i=1
xiei = 3∑
i,j=1
ajixi, i = 1, 2, 3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 4/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Two coordinate systems — rotated IV
A vector u is a geometric quantity with components (u1, u2, u3) in the system (e1, e2, e3), which are related to the components (u′1, u
′ 2, u ′ 3) in the system (e′1, e
′ 2, e ′ 3) in
u′i = 3∑
aijuj i = 1, 2, 3
or expressed as column vectors and standard matrix multiplicationu′1
u′2 u′3
u1 u2 u3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 5/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
A general rotation can be made in three steps (Euler)
e1 e2e ′ 1
′′′ 3 ), are related to each other by
the three Euler angles (α, β, γ) [4, 8] 1 A rotation with the angle α around the e3 axis 2 A rotation with the angle β around the e′2 axis 3 A rotation with the angle γ around the e′′3 axis
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 6/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Euler angles II
=
0 0 1
x1 x2 x3
x′′2 x′′3
sinβ 0 cosβ
The third rotation is represented byx′′′1
x′′′2 x′′′3
cos γ sin γ 0 − sin γ cos γ 0
0 0 1
x′′1 x′′2 x′′3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 7/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
In total, the rotation is made by
r′′′ = Rz(γ)Ry(β)Rz(α)r
The matrices are all orthogonal, R−1 = Rt (corresponds to a change in sign of the angle, e.g., α→ −α)
x′′′1 x′′′2 x′′′3
=
cos γ sin γ 0 − sin γ cos γ 0
0 0 1
sinβ 0 cosβ
0 0 1
x1 x2 x3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 8/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
α
β
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 9/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Two coordinate frames are related by
r′ = Rr
A scalar function f (r) in the unprimed system is related to the function f ′(r′) in the primed system
f ′(r′) def = f (r)
“Same values at the same point in space”
Define an operation PR on scalar functions as f ′ = PRf
(PRf )(Rr) def = f (r)
R rotates coordinates r to r′
PR “rotates” functions f to f ′
Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 10/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Rotation of scalar functions II
Successive rotations, first by R, then by S, i.e., r′′ = Sr′ = SRr leads to
(PSf ′)(Sr′) def = f ′(r′) = f (r) def
= (PSRf )(r′′)
We conclude PSR = PSPR
In particular
I = PI = PR−1R = PR−1PR ⇒ P−1 R = PR−1
Notice that the identity I is used both as the identity operator (for functions) and the identity matrix (for rotations)
Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 11/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
and
(PR(fg))(r) = f (R−1r)g(R−1r) = (PRf )(r)(PRg)(r)
Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 12/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Invariance principles — general I
If A(r) is an operator acting on f (r), giving a new function g(r) = A(r)f (r), then
(PR(Af ))(r) = (PRg)(r) = g(R−1r) = A(R−1r)f (R−1r)
On the other hand
f )(r) = (PRA(r)P−1 R f )(R−1r)
If the operator A satisfies
A(r) = PRA(r)P−1 R ⇔ A(r)PR = PRA(r) (Commutes)
it is called symmetric or invariant under the transformation R — A commutes with PR (in our case symmetric under rotations, e.g., A(r) = ∇2) [5, 10]
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 13/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Invariance principles — general II
We now study what happens to the eigenfunctions of an operator A
Let ψn(r) be eigenfunctions to A(r) with N (distinct) eigenvalues λn, i.e.,
A(r)ψn(r) = λnψn(r), n = 1, 2, . . . ,N
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 14/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Invariance principles — general III
If the operator A is symmetric under the transformation R, the rotated functions, PRψn, are also eigenfunctions of A with the same eigenvalue
A(r)(PRψn)(r) = (PRA(r)ψn)(r) = λn(PRψn)(r), n = 1, 2, . . . ,N
PRψn can be expanded in the set {ψn}N n=1 (linearly
independent set of functions)
(notice the same argument r)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 15/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
PSPRψn = PS
n′′=1
The matrices D(R) represents the group of rotations Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 16/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
or D(R)−1 = D(R−1)
Notice that the identity I is used both as the identity operator for the matrices D(R) and as the identity matrix for rotations in R3
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 17/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
(PRψn)(Rr) def = ψn(r), (definition of PR)
and
= (PR−1ψn)(r) def =
N∑ n′=1
Dn′n(R−1)ψn′(r)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 18/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Representations IV
If the operator A is invariant under the transformation R, then PR are unitary operators, i.e., P†R = P−1
R (the adjoint operator, denoted by a dagger †, is the inverse operator)
f , g = PRf ,PRg = ⟨
P†RPRf , g ⟩
All “observations” in the original and the rotated system must be identical (otherwise there is a special coordinate system, which violated the invariance)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 19/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
If ψn are orthogonal, the PRψn are also orthogonal (PR
unitary operators), i.e.,
δnn′ = PRψn,PRψn′
n′′=1
and the matrices are unitary matrices, D(R)−1 = D(R)†
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 20/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
and
n′=1
Dn′n(R)ψn′(Rr)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 21/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Spherical harmonics I
Let the operator A be the Laplace operator on the unit sphere ∇2
, and we then have N = 2l + 1 eigenfunctions ψn, i.e., {Ylm(θ, φ)}l
m=−l
where
∇2f (r, θ, φ) = 1 r ∂2(rf (r, θ, φ))
∂r2 + 1 r2∇
2 f (r, θ, φ)
The rotation is made with the Euler angles (α, β, γ) [4]
Ylm(θ, φ) = ∑ m′
′, φ′)
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 22/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Assignment
Spherical harmonics II 1 A rotation with the angle α around the e3 axis
Ylm(θ, φ) = eimαYlm(θ′, φ′)
2 A rotation with the angle β around the e′2 axis
Ylm(θ′, φ′) = l∑
′′, φ′′)
3 A rotation with the angle γ around the e′′3 axis
Ylm(θ′′, φ′′) = eimγYlm(θ′′, φ′′′)
m′m(β)eimα
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 23/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
d(l) m′m(β) =
P(m′−m,m′+m) l−m′ (cosβ)
where P(a,b) l (x) are the Jabobi polynomials [1, 4, 7]
1Strictly speaking, m′ − m and m′ + m has to be non-negative Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 24/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
dn
dxn
) P(α,β)
(α− β)
All authors agree on the normalization!!
Special case: If α = β = 0 then P(α,β) n (x) are the
Legendre polynomials Pn(x)
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 25/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
mm′(β)
m0(β)eimγ = (−1)m
P(m,m) l−m (x) =
(1− x2)−m/2Pm l (x)
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 26/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
jl(kr)
vlm′(kr′) ulm′(kr′) wlm′(kr′)
Note that the distance to the origin r is the same in both the primed and the unprimed system
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 27/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Spherical vector waves I
The position vector transforms as (between two rectilinear systems)x′1
x′2 x′3
x1 x2 x3

aij ∂
∂xj
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 28/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Spherical vector waves II
Of particular interest is (somewhat harder to prove, involves the Levi-Civita symbol εijk, permutation symbol [2])
∇′ × (r′f (r)) = ∇× (rf (r))
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 29/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
=
1√ l(l + 1)
= ∑
Dn′n(α, β, γ) = δττ ′δll′D
(l) m′m(α, β, γ) = δττ ′δll′eim′γd(l)
m′m(β)eimα
Note vector identity — primed vectors w.r.t. the primed spherical unit vectors r′, θ
′ , φ ′
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 30/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
φ1(γ; kr) =
β = −x sinβ + y cosβ
γ = x sinα cosβ + y sinα sinβ + z cosα
γ transforms as a vector, but α and β do not
Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 31/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
where
Rττ ′ =
cos −i sin 0 −i sin cos 0
0 0 1
sinα′
sinα′
cosα′ = cosα cos η + sinα sin η cos(β − ψ)
Note vector identity Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 32/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
ψ
η
Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 33/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Cylindrical vector waves
Rotation of cylindrical vector wave is less often employed due to an extended singularity along the e3 axis (radiating waves)
The only one used is rotation of the regular cylindrical vector waves
The rotation of the regular cylindrical vector waves is done by first transforming into planar spherical vector waves (Lecture 7), then a rotation followed by a transformation back
Cylindrical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 34/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
List of references I
[1] M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions. Applied Mathematics Series No. 55. National Bureau of Standards, Washington D.C., 1970.
[2] G. B. Arfken and H. J. Weber. Mathematical methods for physicists. Academic Press, New York, sixth edition, 2005.
[3] L. Brillouin. Tensors in mechanics and elasticity. Academic Press, New York, 1964.
[4] A. R. Edmonds. Angular Momentum in Quantum Mechanics. Princeton University Press, Princeton, New Jersey, second edition, 1960.
[5] M. Hammermesh. Group Theory. Addison-Wesley, Reading, MA, USA, second edition, 1964.
[6] J. E. Hansen, editor. Spherical Near-Field Antenna Measurements. Number 26 in IEE electromagnetic waves series. Peter Peregrinus Ltd., Stevenage, UK, 1988. ISBN: 0-86341-110-X.
[7] G. Kristensson. Second order differential equations — Special functions and their classification. Springer-Verlag, London, 2010.
[8] G. Kristensson. Spherical vector waves, 2011. This booklet is an excerpt of the textbook Gerhard Kristensson, Scattering of Electromagnetic Waves.
List of references Lecture 5: Rotations of Vector Waves December 1, 2011 Page 35/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
List of references II
[9] J. Kuipers. Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual reality. John Wiley & Sons, New Jersey, 2002.
[10] E. Wigner and J. Griffin. Group theory and its application to the quantum mechanics of atomic spectra, volume 4. Academic Press, New York, 1959.
List of references Lecture 5: Rotations of Vector Waves December 1, 2011 Page 36/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Assignment 5
1 Write explicitly down the transformation of the coordinate system in terms of the Euler angles α, β, and γ = 0, and identify the transformed spherical angles θ′ and φ′ in terms of the unprimed θ and φ
2 Write down the Wigner’s functions, d(l) m′m(β) and
D(l) m′m(α, β, 0), for l = 0 and l = 1
3 Verify the transformation of the spherical harmonics
Y00(r) =
√ 1

Y10(r) =
√ 3
for α, β, and γ = 0 — compare with Item 1
Assignment Lecture 5: Rotations of Vector Waves December 1, 2011 Page 37/(38)
Overview
Rotation of scalar functions
Invariance principles — general Representations
mm′ (α)
Assignment
Hint
Aim at expressing the spherical angles for the primed coordinates in the following form
x′3 = r cos θ′, x′1 ± ix′2 = r sin θ′e±iφ′
Assignment Lecture 5: Rotations of Vector Waves December 1, 2011 Page 38/(38)
Rotations of the coordinate system
Rotation of scalar functions