016 Rigid Body Motion Introduction - George Mason University

PHYS 705: Classical MechanicsRigid Body MotionIntroduction + Math Review

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- there are of the constraint equations:

How to describe a rigid body?

Rigid Body - a system of point particles fixed in space

subject to a holonomic constraint:

ij ijr c for all i, j pairs

Question: How many independent coordinates are needed to specify the system?

11

2N N

Let count them:

ijri

j

- N particles at most 3N degrees of freedom

ij ijr c

- For N large, but many equations are not independent ! 11

2N N N

From first inspection, it is not clear how many independent coordinates are there.

2


A better way to think about this…

2dANY point i within the rigid body can be

located by the distances from

three fixed reference points !

3d1d

i

2

3

1 1 2 3, ,d d d

So, we only need to specify the coordinates of the

three reference points; then ALL points in the

rigid body are fixed by the constraint equations,

at most 9 dofs

ij ijr c

3

23r

3

13r


Again, NOT ALL 9 coordinates for the 3 ref pts are independent !

Let count...

Total 6 degrees of freedom

212r

4

1

- 3 coordinates needed to specify the 1st point

(Since point #2 can be anywhere on the

surface of a sphere (fixed distance) centered

on the 1st point.)

(Since point #3 must lie on a circle with

fixed distances to the first two points.)

- 2 more coordinates to fix the 2nd point

- 1 more coordinate to fix the 3rd point


Thus, a rigid body needs only six independent generalized coordinates to

specify its configurations.

Independent on how many particles… even in the

limit of a continuous body.

The next question is: How should one specify these 6 coordinates?

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“Fixed” and “Body” Axes for a rigid body

We will use two sets of coordinates:

- 1 set of external “fixed” (lab) coordinates (unprimed)

- 1 set of internal “body” coordinates (primed)

x

y

z

'x

'y

'z

, ,x y z

', ', 'x y z

(As the name implied, the “body” axes are

attached to the rigid body.)

oo

3 coords to specify the origin o

of the “body” axes wrt the fixed

system

3 coords to specify the orientation of

the “body” axes wrt to the translated

“fixed” system (dotted axes).

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Orientation of the “Body” Axes

There are many ways to describe the orientation of the “body” axes…

- A sequence of 3 rotations in a standard order to get from

the fixed axes (shifted) to the body axes.

- This gives a choice for the 3 orientation coordinates

Most often used: Euler’s angles (later) but similar to (roll, pitch, yaw)

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Orientation of the “Body” Axes

There are many ways to describe the orientation of the “body” axes…

Alternatively, one can use 2 coordinates to specify the orientation of the

axis of rotation (direction points along) and the 3rd coordinate to

specify the angle of rotation around the axis of rotation.

8

ˆ 'z

ˆ 'z

Euler’s and Chasles's Theorems

Euler’s Theorem:

A general displacement of a rigid body with 1 pt fixed in space is

equivalent to a rotation about some axis.

A general displacement = a translation + a rotation

Useful general principle for the analysis of rigid bodies…

Chasles’s Theorem:

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Math Review: Coordinate Transformation

- Specifically, one’s choice of the coordinate system shouldn’t matter

the description of motion should be the same in any coords we choose.

Simple quantities like m are called scalars.

- For an actual physical object in nature, the intrinsic properties of the

object should not depend on how we chose to describe it !

- If there is a transformation linking two coordinate systems, physical

properties should be unaffected by the transformation operation.

e.g., mass m makes no difference to a particular choice of coords

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Invariance and Coordinate Transformation

Different from a scalar, the actual description of

a vector depends on the choice of the coord

system, i.e., the actual components.

Next, more complex quantities : vectors (a set of n numbers )

x

y

z

'x

'y

'z

A A

BUT, there exists a well-defined transformation

for vectors such that …

Vector equation such as F = ma (physical laws)

remain unchanged in different reference frames !

Scalars, Vectors, and more complicated objects such as: Matrices and

Tensors used in defining physical quantities are constrained by how

they remain invariant under a coordinate transformation.

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From now on, we will interchangeably

label our axes with indices, i.e.,

x

y

z

'x

'y

'z

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Notation

1 2 3, , , ,x y z x x x

1 2 3', ', ' ' , ' , 'x y z x x x

1 'x

2 'x

RotationTo describe the orientation of a rigid body, the transformation of interest is:

Rotation

Going from specifying a point P

in unprimed system to its

specification in primed system

1x

2x

P

1x

2x

Relating from 1'x 1 2,x x

2green line = sinx 1 = red lin se cox

1' + = ered gre nx

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1'x

1 2cos sinx x

Rotation

1x

2x

1 'x

2 'xP

1x

2x

Similarly, relating from :2'x 1 2,x x

1green line = sinx

2 = red lin se cox 1'x

2 2 1 1 2' = cos sin sire gd n coe sre nx x x x x

1x

2'x

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RotationThese two transformation equations can be put in a compact matrix form:

1 1 1

2 2 2

' cos sin

' sin cos

x x x

x x x

R

R is called a rotation matrix and it takes P from the unprimed frame to the

prime frame.

Putting all entries in terms of cosine, we have:

1 1

2 2

cos cos 2'

cos 2 cos'

x x

x x

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RotationThis is a particular elegant way to express the rotation matrix

where are called the directional cosines and it is the cosine of the angle

between the corresponding axes !

i.e.,

1 1 11 12 1

2 2 21 22 2

cos cos 2'

cos 2 cos'

x x a a x

x x a a x

ija

cosine of angle between ' and axesij i ja x x

11a

2x

1'x2x

2

12a 1'x

2'x

21a 1x

2

2'x

1x

22a

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Rotation- Throughout our discussion, the point

P is “fixed” and the prime frame rotates

counter-clockwise

This is the “passive” point of view.1x

2x

1 'x

2 'xP

- One could equally consider the

transformation (rotation) as taking the

point P (or vector) and rotating it by in

the clockwise direction in the same frame.

1x

2x

P

'P

This is the “active” point of view.

Either way, the math is the same !

(convention: + counterclockwise)

(convention: + clockwise)

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i.e., to rotate about the axis:

Rotation

- In 3 dimensions, it is particular simple to get the matrix for a rotation about

one of the coordinate axes:

Just put a “1” in the diagonal corresponding to that axis and “squeeze”

the 2D rotation matrix into the rest of the entries .

i.e., to rotate about the axis:3x

2x

cos sin

sin cos

0

0

0 0 1

0

0 1 0

cos sin

sin cos0

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Rotation

Examples:

1x

2x

3x

1'x2'x

3'x

900 1 0

1

1

0 0

0 0

use

1x

2x

3x

1'x

2'x

3'x

90 0 0

0 0 1

0 0

1

1

use

(recall: + is counter-clockwise in passive view)

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RotationTo do compound rotations, the matrices multiply together but need to

pay attention to the order of multiplication.

1x

2x

3x

1'x2'x

3'x

90

1

0 1 0

1 0 0

0 0

'

x x

1''x

2''x

3''x

90 0 0

0 0 1

0 1 0

'

1

' '

x x

1st

Rotation:

1'x2'x

3'x2nd

Rotation:

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Rotation

Combining the two rotations:

1x

2x

3x

1'x

2'x

3'x

901''x

2''x

3''x

1

1

0 0 0 1 0 0 1 0

0 0 1 1 0 0 0 0 1

0 1 0 0 0 1 0 0

''

xx x

1st2nd

IMPORTANT : Rotations do not commute reverse order gives different results.

90first

then …

1x

3'x

0 1 0 0 0 0 0 1

1 0 0 0 0 1 1 0 0

0 0 0 1 0 0

'

1

'

1 1 0

xx x NOT the same

as before !

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Euler’s Angles consisting of a particular sequence of 3

rotations (D, C, B) along three principle axes:

Euler’s Angles

- However, a general rotation about an arbitrary axis is not quite simple. In

general all entries will be nonzero !

, ,

- But, one can build up a general rotation as separate successive rotations.

- There are many conventions... We will choose one called the

B

'

C

'

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D

The first rotation D is about by an angle :

Euler’s Angles

Start with the “fixed” axes , ,x y z

The second rotation C is about the new

(new x-axis) by an angle :

D C B

z cos sin 0

sin co

1

s 0

0 0

D

0 0

0 cos sin

0 sin c

1

os

C

The third rotation B is about the new (new

z-axis) by an angle :

'

( ) , , , ,space axes x y z

, , ', ', '

cos sin 0

sin co

1

s 0

0 0

B

' '

', ', ' ', ', ' ( )x y z body axes

demo: http://demonstrations.wolfram.com/EulerAngles/

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Euler’s Angles

- Then the general rotation from the “fixed” axes to the “body”

axes is given by the product:

, ,x y z

(The sequence of rotated angles are called the Euler’s angles.)

D C B

( , , ) ( ) ( ) ( ) A B C D

, ,

cos cos cos sin sin cos sin cos cos sin sin sin

sin cos cos sin cos sin sin cos cos cos sin cos

sin sin sin cos cos

A

' '

', ', 'x y z

' ( )body fixedx Ax

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(note the order of operations !)

For a general transformations, the nine matrix elements,

Back to a bit of Math Review: Vectors & Matrices

Write a general matrix multiplication out:

3

1

' 'j ji ij

x a x

x Ax- The ith component of is:

where are the ijth component (ith row, jth column) of the

transformation matrix A. Note,

ija

, 1,2,3 and 1,2,3ija i j

But, for rotations, they will not be all independent since the transformation

of vectors must satisfy the property that

the length of a vector must remain invariant

can be unrelated parameters

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'x

Tj ji ia a A A

A bit of Math Review: Vectors & Matrices

- In the unprimed frame, let

Vector with the same magnitude

1 2 3, ,x x xxRecall the picture for vectors under rotations…

- In the primed frame, we have 1 2 3' ' , ' , 'x x xx

1'x

2'x

1x

2x

2 22 2' ' iii i

x x x x

This gives the following condition:

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Putting in the transformation equation for explicitly, we have,

For the RHS to be equal to

'x

2

2' ij j ij j ik ki

i i j i j k

x a x a x a x

We need to have the following condition:

ij ik j ki j k

a a x x 22

ii

x x

0

1ij ik jki

for j ka a

for j k

“orthogonality condition”

where is the kronecker delta.jk

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jij ik iki i

ia a a a NOTE:


Comments:

1. To simplify the summation signs, we will use the Einstein convention

(sum over repeated indices) , i.e.,

2. Although we have 3 x 3 = 9 of the orthogonality conditions for

2. ., i i ii

e g x x xand we have the orthogonality condition: i sums overj ki ji ka a

, 1, 2,3j k

But, since are just numbers and (commute), we only have

one equation for each unique jk pair and this gives six independent

conditions (pairs of jk).

i j ik ik i ja a a aija

9 and 6 conditions 3 free parameters (3 Euler’s angles)ija

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In Einstein’s notation, we can rewrite the transformation/rotation as:

A vector is a rank-1 tensor meaning that we need only one to transform it.

'i ij jx a x with ij ik jka a

(more on tensors later)

ija

We will define a rank-2 tensor as an object that transform according to:

'ij ik jl klI a a Iwith similar orthogonality conditions on

both so that tensor equations

involving will be invariant.

ik jla and a

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ijI

Now, we will review some general notations and properties of matrices and

transformations:


Tik kj ki jk jk kia b a b b a

Tij jia a

ij ik kjc a b

1. Matrix Multiplication: C AB

(sum over k)

2. Multiplication does not commute: AB BA

summing index diff

3. Multiplication is associative: AB C A BC

4. is the transposeTA

T T TAB B A

5. is the inverse, where 1

1

1

0

0

AA I 1A 1ij ikjk

a a or

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ik kj kj ik jk kia b b a b a 1st index 2nd index

ik kj jl ik kj jla b c a b c

Continuing…


6. For an orthogonal matrix, we have : 1 T A A

(sum over i)7. Trace: iiTr aA

8. Determinant:

11 1

1

1

minor n

k

m mn

a a

a a

A

det 1a for n A

1

1 1det 1 det minor 1k

k kk

a for n A A

1jiij

a a

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example:


deta b

a d b c ad bcc d

A

11 12 1322 23 21 23 21 22

21 22 23 11 12 1332 33 31 33 31 23

31 32 33

a a aa a a a a a

a a a a a aa a a a a a

a a a

(can be expanded in any row or column)

9. Other properties of det:

1 1det

det A

A det detT A A det det detAB A B

For an orthogonal matrix,

21 det det det detT T AA I AA I A A A

So, if A is an orthogonal matrices!det 1 A

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ijk

10. Levi Civita Tensors:

1, is of forward permutation

1, is of backward permutation

0, otherwise ijk

ijk

ijk

(123, 231, 312)

(321, 213, 132)

1 2 3det ijk i j ka a aA

In terms of …

i ijk j kc a b C A B

e.g., 1 123 2 3 132 3 2 2 3 3 2c a b a b a b a b

A useful identity:

ijk lmk il jm im jl

(sum over k) (1st & 2nd match) (mixed)

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ijkExample using

ijk j k ilm l ma b c d

A B C D

(they are numbers…order does not matter)

(more exercise in homework)

(permuting does not change the sum)

(dot product sums over i)

(use pervious identity)

(collapse delta function )

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l m l m m l l ma b c d a b c d

ijk ilm j k l ma b c d

jki lmi j k l ma b c d

jl km jm kl j k l ma b c d

A C B D A D B C

11. Given a matrix , is an eigenvector with eigenvalue l

A bit of Math Review: Eigenvalues & Eigenvectors

A

lAx xif

nx

- Thinking in terms of active transformations, a vector is an eigenvector

of a transformation A if A merely stretch it by a factor l.

12. Eigenvalues can be founded by solving the characteristic equation of the

matrix A:

det 0l A I (since A is n dimensional, in general, there will be n roots)

13. 1 2det nl l lA (including multiplicity (degeneracy))

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14. All of the following statements are equivalent:

A bit of Math Review: Eigenvalues & Eigenvectors

- A is invertible meaning exists and such that

det 0 A

1A 1 AA I

- of A are linearly independentcolumns

rows

- is invertibleTA

- is nonsingularA

- For a given non-zero vector b, a unique vector x s.t. Ax b

- All eigenvalues 0

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15. Similarity Transformations

A bit of Math Review: Similarity Transformation

- Given an n-dim vector space V, choose a particular basis

In the active sense, a linear transformation A (such as a rotation)

takes a vector x in V into another vector x’ in V

- We would say that A and B represent the “same” transformation or A is

similar to B, i.e.,

- Now, if one choose a different basis

The “same” transformation will in general be represented by a

different matrix, let say B.

a nonsingular matrix C such that

1A C BC (this is called a Similarity Transform)

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Looking at this closer…


- If A and B are the “same”, then we want

(vectors transform under A)

B Cx C Ax

basis #1 basis #2

x

Ax

(vectors transform under B)

Cx

C Ax

1A C BC

(If this holds, then A and B will have the “same” active effects

on vectors.)

C

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B

Important properties of a Similarity Transformation:


also,

1det 1/ det C C 1det det det A C BC B

Sometimes, there is a “best” or “most natural” basis in which

to represent a linear transformation.

since

1 1 1ii jk k jki ijj kTr Tr c b c bc c A C BC

i Tr index

Similarity Transformations are important:

It is advantageous in many cases if the linear

transformation can be expressed as a diagonal matrix.

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k kkj j kb b

Tr

B

For a given symmetric matrix B, there exist a matrix A which is similar to

B such that


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0

0 n

l

l

A C BC A is diagonal and its entries

are the eigenvalues

- All eigenvalues (multiplicity possible) are real

- The columns of C are the eigenvectors of B

For non-symmetric matrices, l’s might be complex. If one is restricted to

real-value matrices, then the best that one can do is that A is block diagonal

(Jordan form) with 2 x 2 blocks corresponding to pairs of complex conjugate

l’s.

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Another use of similarity transformation, recall Euler/Chasles’ Theorem:

Back to Rigid Body (Finite Rotations)

Any general displacement of a rigid body with 1 point fixed is

equivalent to a rotation about a given axis.

So, instead of 3 Euler rotations , one could do

ONE rotation if one know where the axis is located.

, ,

fixed at this point

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There is a formula that relates these 4 angles and we can obtain this equation

by considering the following Similarity Transformation:


cos sin 0

Similarity Transformation sin cos 0

0 0 1

full

Euler matrix

fixed at this point

rotate about by ˆ 'z

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ˆ 'z

In particular, there exists a similar matrix such that:


-BCD are the three Euler rotations

-S is a nonsingular matrix for the Similar Transformation

-R() is the (simple) rotation matrix along the given axis

1 ( ) BCD S R S

cos cos cos sin sin cos sin cos cos sin sin sin

sin cos cos sin cos sin sin cos cos cos sin cos

sin sin sin cos cos

BCD

cos sin 0

sin cos 0

0 0 1

R

ˆ 'z

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S

Since the trace is invariant under a Similarity Transformation, we have


1Tr Tr Tr BCD S RS R

RHS:

cos sin 0

sin cos 0 2 cos 1

0 0 1

Tr Tr

R

cos sin sicos cos

s

n

cos cos co cosin in ss

Tr

BCD

LHS:

cos cos cos

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So, the LHS simplify to,


cos 1 cos cos 2cos 11 1

cos 1 cos cosTr BCD

Now, RHS+1 = LHS +1 gives,

1 cos 1 cos 2 cos 1

2 2cos 22

2cos 22

2 2cos2

cos cos cos2 2 2

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016 Rigid Body Motion Introduction - George Mason University

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