Lecture 8 - Harvard University Department of...

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Mechanics Physics 151 Lecture 8 Rigid Body Motion (Chapter 4)

Transcript of Lecture 8 - Harvard University Department of...

MechanicsPhysics 151

Lecture 8Rigid Body Motion

(Chapter 4)

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

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