Mechanics
Physics 151
Lecture 8
Rigid Body Motion
(Chapter 4)
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Mechanics Rigid Body Motion, Lecture Notes - Physics, Study notes of Mechanics
Mechanics, Physics, Rigid Body Motion, Rigid Body, 2-D Rotation, 3D Rotation, Constraints of Rotation , Orthogonal Matrix, Space Inversion, Rotation Matrix, Euler Angles, Euler’s Theorem, Rotation Vecto,r Infinitesimal Rotation, Axial Vecto,r Parity.
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Mechanics
Physics 151
Lecture 8
Rigid 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
) changes
(^
)^
sin
s^
ds d
hits N
I
=^
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 3
N
and subtract the number of constraints
!^
Right answer: 3 translation and 3 rotation = 6
const
for all ,
ij^
i^
j
r^
i j
=^
−^
r^
r
2
(^
N N
N
N
N
−^
−^
=^
≤^
0 for N
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
sin
sin
cos
x^
x
y^
y x y
θ^
θ^
^
^
^
^
^
^
′^
^
^
^
′^
⋅^
^
=^
^
′^
⋅^
^
i^
i^
i^
j
j^
i^
j^
j
θ^
′ x x
y ′ y
i
′ j
′ i j
x^
′ x
y ′ y
z
′ z
3D Rotation!^
Simplify formulae by renaming^!
Rotation is now expressed by
!^
We got 9 parameters
a
ij^
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
′^
′^
′^
′^
′^
cos
i^
ij^
j^
ij^
j^
ij^
j
j^
j
x^
x^
a x
a x
=^
∑
∑ 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
2
i^
i^
i^
i
r^
x x
′ ′ x x
=^
i^
i^
ij^
j^
ik^
k
x x
a x a x ′ ′ =
i^
ij^
j
x^
a x ′ = therefore
(^
(^
ij^
ik^
jk
j^
k
a a
j^
k
δ^
=^
≡^
^
6 conditionsreduces freeparametersfrom 9 to 3
AA
Transpose of
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 properorthogonal matrices
^
=^
≡^
^
^
^
r^
r^
Sr
r^
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 thesame 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!^
Definition of Euler angles is somewhat arbitrary^!
cos May rotate around different axes in different order! Many conventions exist – Watch out!
sin
0
sin
cos
0
0
0
1
φ^
φ
φ^
φ
^
^
=^
−
^
^
D
1
0
0
0
cos
sin
0
sin
cos θ^
θ
θ^
θ
^
^
=^
^
^
− ^
C
cos
sin
0
sin
cos
0
0
0
1
ψ^
ψ
ψ^
ψ
^
^
=^
−
^
^
B
cos
cos
cos
sin
sin
cos
sin
cos
cos
sin
sin
sin
sin
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^
θ
ψ^
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^
!^
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
t ),
t ),
( t )
!^
A
^1
!^
6 independent coordinates (
x ,
y
,^ z
,^ φ
,^ θ
,^ ψ
Euler’s Theorem
!^
If a matrix
A
satisfies
!^
Since !^
For odd-dimensioned matrices
Ar = r
(^
)^
A
1 r =
0 or
0 or
−^
=^
=^
A
r^
A - 1
A
! A
(^
−^
−^
−^
A
1 A = 1
A
A
1 A = 1
A
A
A
!^
!^
−^
M
M
−^
−^
−^
A
A
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 whichrotation is made first
behaves almost like a
vector
Infinitesimal Rotation!^
Inverse of an infinitesimal rotation is^!
Using! We can write
ε as
(^
+^
−^
=^
+^
−^
ε^
ε^
ε^
ε^
1
(^
+^
=^
ε^
ε (^1) − = A
!^ A
+^
=^
ε^
ε
!
ε^
ε !^
ε^ is antisymmetric^3
2
3
1
2
1
d^
d
d^
d
d^
Ω d
^
^
=^
^
^
^
ε^
1
2
3
(^
,^
,^
d^
d^
d^
d
=^
We’ll see…
Infinitesimal Rotation!^
A vector
r
is rotated by
(^1
ε
)^ as
!^
Euler’s theorem says this equals to arotation by an infinitesimal angle
d
around an axis
n
(^
r^
ε^
r
3
2
1
3
1
2
2
1
3
d^
d^
x
d^
d^
x
d^
d^
x
d^
d
−^
^
^
≡^
−^
=^
=^
^
^
−^
^
r^
r^
r^
ε^
r ◊
r^
n
r d r
d Φ
d^
d
=^
r^
r ◊n
d^
d =^
n