Post on 08-Jun-2018
What We Did Last Time
! Discussed scattering problem! Foundation for all experimental physics
! Defined and calculated cross sections! Differential cross section and impact parameter! Rutherford scattering
! Translated into laboratory system! Angular translation + Jacobian! Shape of σ(Θ) changes
( )sin
s dsd
σ Θ =Θ Θ
hitsN I σ= ⋅
Goals For Today
! Start discussing rigid-body motion! Multi-particle system with “fixed shape”
! Concentrate on representing the rotation! Which generalized coordinates should we use?! Define Euler angles! Define infinitesimal rotation
! Will use this for angular velocities, etc
! Today’s lecture is largely mathematical! Assume knowledge of linear algebra
Rigid Body
! Multi-particle system with fixed distances! Constraints:
! How should we define generalized coordinates?! How many independent coordinates are there?! If you start from 3N and subtract the number of constraints
! Right answer: 3 translation and 3 rotation = 6
const for all ,ij i jr i j= − =r r
2( 1) 732 2
N N N NN − −− = ≤ 0 for N ≥ 7
Not all the constraints are independent
Today’s theme
2-D Rotation
! 2-dimensional rotation is specified by a 2×2 matrix
! Try the same thing with 3-d rotation
cos sinsin cos
x xy y
xy
θ θθ θ
′ = ′ −
′ ′⋅ ⋅ = ′ ′⋅ ⋅
i i i jj i j j
θ x
x′y
y′
i
′j′ij
x x′y
y′z
z′
3D Rotation
! Vector r is represented in x-y-z and x’-y’-z’ as
! Using angles θij between two axesx y z x y z′ ′ ′ ′ ′ ′= + + = + +r i j k i j k
11 12 13cos cos cosx x y z
x y zθ θ θ′ ′ ′ ′ ′= ⋅ = ⋅ + ⋅ + ⋅= + +
r i i i j i k i
21 22 23cos cos cosy x y zθ θ θ′ = + +
31 32 33cos cos cosz x y zθ θ θ′ = + + x x′y
y′z
z′
11θ
12θ13θ
11 12 13
21 22 23
31 32 33
cos cos coscos cos coscos cos cos
x xy yz z
θ θ θθ θ θθ θ θ
′ ′ =
′
or
3D Rotation
! Simplify formulae by renaming
! Rotation is now expressed by
! We got 9 parameters aij to describe a 3-d rotation! Only 3 are independent
1 2 3( , , ) ( , , )x y z x x x→ 1 2 3( , , ) ( , , )x y z x x x′ ′ ′ ′ ′ ′→
cosi ij j ij j ij jj j
x x a x a xθ′ = = =∑ ∑
Einstein convention:Implicit summation over repeated index
Constraints of Rotation
! Rotation cannot change the length of any vector! Exactly the constraints we need for rigid body motion
! Using the transformation matrix
! Matrix A = [aij] is orthogonal
2i i i ir x x x x′ ′= =
i i ij j ik kx x a x a x′ ′ =i ij jx a x′ =
therefore1 ( )0 ( )ij ik jk
j ka a
j kδ
== ≡ ≠
6 conditions reduces free parametersfrom 9 to 3
=AA 1!Transpose of A
Orthogonal Matrix
! Goldstein Section 4.3 covers algebra of matrices! You must have learned this already
! Orthogonal matrix A satisfies! Consider the determinants
! |A| = +1 " proper matrix! |A| = –1 " improper matrix
11 12 13
21 22 23
31 32 33
a a aa a aa a a
=
A
ij ik jka a δ==AA 1!
Transposed matrix2 1= = =AA A A A! ! 1= ±A
Space Inversion
! Space inversion is represented by
! S is orthogonal Doesn’t change distances! But it cannot be a rotation
! Coordinate axes invert to become left-handed! Orthogonal matrices with |A| = –1 does this
! Rigid body rotation is represented by proper orthogonal matrices
1 0 00 1 00 0 1
− ′ = − = ≡ − −
r r Sr r 1= −S
Rotation Matrix
! A operating on r can be interpreted as! Rotating r around an axis by an angle
! Positive angle = clockwise rotation! Rotating the coordinate axes around the same axis by the
same angle in the opposite direction! Positive angle = counter clockwise rotation
! Both interpretations are useful! We are more interested in the latter for now
! How do we write A with 3 parameters?! There are many ways
′ =r Ar
Euler Angles
! Transform x-y-z to x’-y’-z’ in 3 steps( , , )x y z( , , )ξ η ζ( , , )ξ η ζ′ ′ ′
( , , )x y z′ ′ ′
Rotate CCW by φaround z axis
Rotate CCW by θ around ξ axis
Rotate CCW by ψ around ζ’ axis
z
xy
ξ
η
ζ z
xy
ξ ′
η′ζ ′
φ
θz
xy
z′
x′
y′
ψ
x
DxCDx
=Ax BCDx
Euler Angles
! Definition of Euler angles is somewhat arbitrary! May rotate around different axes in different order! Many conventions exist – Watch out!
cos sin 0sin cos 00 0 1
φ φφ φ
= −
D1 0 00 cos sin0 sin cos
θ θθ θ
= −
Ccos sin 0sin cos 00 0 1
ψ ψψ ψ
= −
B
cos cos cos sin sin cos sin cos cos sin sin sinsin cos cos sin cos sin sin cos cos cos cos sin
sin sin sin cos cos
ψ φ θ φ ψ ψ φ θ φ ψ ψ θψ φ θ φ ψ ψ φ θ φ ψ ψ θ
θ φ θ φ θ
− + = − − − + −
A
Rigid Body Motion
! Motion of a rigid body can be described by:! Define x’-y’-z’ axes (body axes) attached to the rigid body
! Same direction as x-y-z (space axes) at t = 0! Origin fixed at one point of the rigid body (e.g. CoM)
! Use R(t) to describe the motion of the origin! Use A(t) to describe the rotation of the x’-y’-z’ axes
! Use Euler angles φ(t), θ(t), ψ(t)! A(0) = 1 " φ(0) = θ(0) = ψ(0) = 0
! 6 independent coordinates (x, y, z, φ, θ, ψ)
Euler’s Theorem
! In other words! Arbitrary 3-d rotation equals to one rotation around an axis! Any 3-d rotation leaves one vector unchanged
! For any rotation matrix A! There exists a vector r that satisfies! A has an eigenvalue of 1
The general displacement of a rigid body with one point fixed is a rotation about some axis
Ar = rEigenvector with
eigenvalue 1
Euler’s Theorem
! If a matrix A satisfies
! Since
! For odd-dimensioned matrices
Ar = r( ) 0−A 1 r = 0 or 0 or 0− = = =A 1 r A -1
1− =A A!
( )− −
− −
− −
A 1 A = 1 A
A 1 A = 1 A
A 1 = 1 A
! !
! !
− = −M M0− − − =A 1 = A 1
Q.E.D.
Rotation Vector?
! Euler’s theorem provides another way of describing3-d rotation! Direction of axis (2 parameters) and angle of rotation (1)! It sounds a bit like angular momentum
! Critical difference: commutativity! Angular momentum is a vector
! Two angular momenta can be added in any order! Rotation is not a vector
! Two rotations add up differently depending on which rotation is made first
Infinitesimal Rotation
! Small (infinitesimal) rotations are commutative! They can be represented by vectors! We also need them to describe how a rigid body changes
orientation with time
! Infinitesimal rotation must be close to non-rotation
! Two successive infinitesimal rotations make
! Obviously commutative
i i ij jx x xε′ = + ( )′ = +x 1 ε xor 1ijε "
1 2 1 2 1 2
1 2
( )( )+ + = + + += + +
1 ε 1 ε 1 ε ε ε ε1 ε ε
2nd order of ε vanishes
behaves almost like a vector
Infinitesimal Rotation
! Inverse of an infinitesimal rotation is
! Using
! We can write ε as
( )( )+ − = + − =1 ε 1 ε 1 ε ε 11( )−+ = −1 ε 1 ε1− =A A! + = −1 ε 1 ε!
= −ε ε! ε is antisymmetric
3 2
3 1
2 1
00
0
d dd d
d d
Ω − Ω = − Ω Ω Ω − Ω
ε1 2 3( , , )d d d d= Ω Ω ΩΩ
We’ll see…
Infinitesimal Rotation
! A vector r is rotated by (1 + ε) as
! Euler’s theorem says this equals to a rotation by an infinitesimal angle dΦaround an axis n
( )′ = +r 1 ε r
3 2 1
3 1 2
2 1 3
00
0
d d xd d x
d d xd d
Ω − Ω ′≡ − = = − Ω Ω = Ω − Ω
r r r ε r ×r Ω
n
rdr
dΦ
d d= Φr r ×nd d= ΦΩ n
Axial Vector
! dΩ behaves pretty much like a vector! dΩ rotates the same way as r with coordinate rotations
! Space inversion S reveals difference! Ordinary vector flips! dΩ doesn’t
! Such a “vector” is called an axial vector! Examples: angular momentum, magnetic field
′ = = −r Sr r( )d d
d d d′ ′ ′= ×
′= − = − × = ×r r Ω
r r Ω r Ω
d d′ =Ω Ω
Parity
! Parity operator P represents space inversion
( , , ) ( , , )x y z x y z → − − −P
EigenvalueParityQuantity
+1PV* = V*Axial vector−1PV = −VVector−1PS* = −S*Pseudoscalar+1PS = SScalar
*V × V = V* *S⋅V V =
* *S V = V* *S V = V* * S⋅V V =
*V × V = Vetc.
Summary
! Discussed 3-dimensional rotation! Preparation for rigid body motion
! Movement in 3-d + Rotation in 3-d = 6 coordinates
! Looked for ways to describe 3-d rotation! Euler angles one of the many possibilities! Euler’s theorem
! Defined infinitesimal rotation dΩ! Commutative (unlike finite rotation)! Behaves as an axial vector (like angular momentum)
! Ready to go back to physics